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<span>Knot equivalence</span> </div> </a> <ul id="toc-Knot_equivalence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Knot_diagrams" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Knot_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Knot diagrams</span> </div> </a> <button aria-controls="toc-Knot_diagrams-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Knot diagrams subsection</span> </button> <ul id="toc-Knot_diagrams-sublist" class="vector-toc-list"> <li id="toc-Reidemeister_moves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reidemeister_moves"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Reidemeister moves</span> </div> </a> <ul id="toc-Reidemeister_moves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Knot_invariants" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Knot_invariants"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Knot invariants</span> </div> </a> <button aria-controls="toc-Knot_invariants-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Knot invariants subsection</span> </button> <ul id="toc-Knot_invariants-sublist" class="vector-toc-list"> <li id="toc-Knot_polynomials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Knot_polynomials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Knot polynomials</span> </div> </a> <ul id="toc-Knot_polynomials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperbolic_invariants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hyperbolic_invariants"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Hyperbolic invariants</span> </div> </a> <ul id="toc-Hyperbolic_invariants-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Higher_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Higher dimensions</span> </div> </a> <button aria-controls="toc-Higher_dimensions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Higher dimensions subsection</span> </button> <ul id="toc-Higher_dimensions-sublist" class="vector-toc-list"> <li id="toc-Knotting_spheres_of_higher_dimension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Knotting_spheres_of_higher_dimension"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Knotting spheres of higher dimension</span> </div> </a> <ul id="toc-Knotting_spheres_of_higher_dimension-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Adding_knots" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Adding_knots"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Adding knots</span> </div> </a> <ul id="toc-Adding_knots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tabulating_knots" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tabulating_knots"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Tabulating knots</span> </div> </a> <button aria-controls="toc-Tabulating_knots-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Tabulating knots subsection</span> </button> <ul id="toc-Tabulating_knots-sublist" class="vector-toc-list"> <li id="toc-Alexander–Briggs_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alexander–Briggs_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Alexander–Briggs notation</span> </div> </a> <ul id="toc-Alexander–Briggs_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dowker–Thistlethwaite_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dowker–Thistlethwaite_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Dowker–Thistlethwaite notation</span> </div> </a> <ul id="toc-Dowker–Thistlethwaite_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conway_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conway_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Conway notation</span> </div> </a> <ul id="toc-Conway_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gauss_code" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gauss_code"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Gauss code</span> </div> </a> <ul id="toc-Gauss_code-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further reading subsection</span> </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Introductory_textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introductory_textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Introductory textbooks</span> </div> </a> <ul id="toc-Introductory_textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surveys" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surveys"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Surveys</span> </div> </a> <ul id="toc-Surveys-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-History_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>History</span> </div> </a> <ul id="toc-History_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Knot_tables_and_software" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Knot_tables_and_software"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Knot tables and software</span> </div> </a> <ul id="toc-Knot_tables_and_software-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input 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type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 37 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-37" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">37 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%B9%D9%82%D8%AF_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="نظرية العقد (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="نظرية العقد (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_nuedos" title="Teoría de nuedos – Asturian" lang="ast" hreflang="ast" data-title="Teoría de nuedos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_nusos" title="Teoria de nusos – Catalan" lang="ca" hreflang="ca" data-title="Teoria de nusos" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_uzl%C5%AF" title="Teorie uzlů – Czech" lang="cs" hreflang="cs" data-title="Teorie uzlů" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Knudeteori" title="Knudeteori – Danish" lang="da" hreflang="da" data-title="Knudeteori" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Knotentheorie" title="Knotentheorie – German" lang="de" hreflang="de" data-title="Knotentheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%BA%CF%8C%CE%BC%CE%B2%CF%89%CE%BD" title="Θεωρία κόμβων – Greek" lang="el" hreflang="el" data-title="Θεωρία κόμβων" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_nudos" title="Teoría de nudos – Spanish" lang="es" hreflang="es" data-title="Teoría de nudos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Korapiloen_teoria" title="Korapiloen teoria – Basque" lang="eu" hreflang="eu" data-title="Korapiloen teoria" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%AF%D8%B1%D9%87%E2%80%8C%D9%87%D8%A7" title="نظریه گره‌ها – Persian" lang="fa" hreflang="fa" data-title="نظریه گره‌ها" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_n%C5%93uds" title="Théorie des nœuds – French" lang="fr" hreflang="fr" data-title="Théorie des nœuds" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_de_n%C3%B3s" title="Teoría de nós – Galician" lang="gl" hreflang="gl" data-title="Teoría de nós" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A7%A4%EB%93%AD_%EC%9D%B4%EB%A1%A0" title="매듭 이론 – Korean" lang="ko" hreflang="ko" data-title="매듭 이론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hn%C3%BAtafr%C3%A6%C3%B0i" title="Hnútafræði – Icelandic" lang="is" hreflang="is" data-title="Hnútafræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_dei_nodi" title="Teoria dei nodi – Italian" lang="it" hreflang="it" data-title="Teoria dei nodi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%A9%D7%A8%D7%99%D7%9D" title="תורת הקשרים – Hebrew" lang="he" hreflang="he" data-title="תורת הקשרים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%BE%E0%A4%A0_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="गाठ सिद्धान्त – Marathi" lang="mr" hreflang="mr" data-title="गाठ सिद्धान्त" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_simpulan" title="Teori simpulan – Malay" lang="ms" hreflang="ms" data-title="Teori simpulan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Knopentheorie" title="Knopentheorie – Dutch" lang="nl" hreflang="nl" data-title="Knopentheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B5%90%E3%81%B3%E7%9B%AE%E7%90%86%E8%AB%96" title="結び目理論 – Japanese" lang="ja" hreflang="ja" data-title="結び目理論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Knuteteori" title="Knuteteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Knuteteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%B1%E0%A8%A0_%E0%A8%B8%E0%A8%BF%E0%A8%A7%E0%A8%BE%E0%A8%82%E0%A8%A4" title="ਗੱਠ ਸਿਧਾਂਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੱਠ ਸਿਧਾਂਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_w%C4%99z%C5%82%C3%B3w" title="Teoria węzłów – Polish" lang="pl" hreflang="pl" data-title="Teoria węzłów" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_dos_n%C3%B3s" title="Teoria dos nós – Portuguese" lang="pt" hreflang="pt" data-title="Teoria dos nós" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_nodurilor" title="Teoria nodurilor – Romanian" lang="ro" hreflang="ro" data-title="Teoria nodurilor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%83%D0%B7%D0%BB%D0%BE%D0%B2" title="Теория узлов – Russian" lang="ru" hreflang="ru" data-title="Теория узлов" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Knot_theory" title="Knot theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Knot theory" 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searchaux" style="display:none">Study of mathematical knots</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tabela_de_n%C3%B3s_matem%C3%A1ticos_01,_crop.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Tabela_de_n%C3%B3s_matem%C3%A1ticos_01%2C_crop.jpg/220px-Tabela_de_n%C3%B3s_matem%C3%A1ticos_01%2C_crop.jpg" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Tabela_de_n%C3%B3s_matem%C3%A1ticos_01%2C_crop.jpg/330px-Tabela_de_n%C3%B3s_matem%C3%A1ticos_01%2C_crop.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Tabela_de_n%C3%B3s_matem%C3%A1ticos_01%2C_crop.jpg/440px-Tabela_de_n%C3%B3s_matem%C3%A1ticos_01%2C_crop.jpg 2x" data-file-width="3309" data-file-height="3294" /></a><figcaption>Examples of different knots including the <a href="/wiki/Trivial_knot" class="mw-redirect" title="Trivial knot">trivial knot</a> (top left) and the <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knot</a> (below it)</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TrefoilKnot_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/220px-TrefoilKnot_01.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/330px-TrefoilKnot_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/440px-TrefoilKnot_01.svg.png 2x" data-file-width="250" data-file-height="250" /></a><figcaption>A knot diagram of the trefoil knot, the simplest non-trivial knot</figcaption></figure> <p>In <a href="/wiki/Topology" title="Topology">topology</a>, <b>knot theory</b> is the study of <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">mathematical knots</a>. While inspired by <a href="/wiki/Knot" title="Knot">knots</a> which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "<a href="/wiki/Unknot" title="Unknot">unknot</a>"). In mathematical language, a knot is an <a href="/wiki/Embedding" title="Embedding">embedding</a> of a <a href="/wiki/Circle" title="Circle">circle</a> in 3-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1f81bfef66c67c01e157738c3bd11dc82a98be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {E} ^{3}}"></span>. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> upon itself (known as an <a href="/wiki/Ambient_isotopy" title="Ambient isotopy">ambient isotopy</a>); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself. </p><p>Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. </p><p>A complete algorithmic solution to this problem exists, which has unknown <a href="/wiki/Computational_complexity" title="Computational complexity">complexity</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In practice, knots are often distinguished using a <i><a href="/wiki/Knot_invariant" title="Knot invariant">knot invariant</a></i>, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include <a href="/wiki/Knot_polynomials" class="mw-redirect" title="Knot polynomials">knot polynomials</a>, <a href="/wiki/Knot_group" title="Knot group">knot groups</a>, and hyperbolic invariants. </p><p>The original motivation for the founders of knot theory was to create a table of knots and <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">links</a>, which are knots of several components entangled with each other. More than six billion knots and links <a href="/wiki/Knot_tabulation" title="Knot tabulation">have been tabulated</a> since the beginnings of knot theory in the 19th century. </p><p>To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other <a href="/wiki/3-manifold" title="3-manifold">three-dimensional spaces</a> and objects other than circles can be used; see <i><a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot (mathematics)</a></i>. For example, a higher-dimensional knot is an <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-dimensional sphere</a> embedded in (<i>n</i>+2)-dimensional Euclidean space. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_knot_theory" title="History of knot theory">History of knot theory</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:KellsFol034rXRhoDet3.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/KellsFol034rXRhoDet3.jpeg/190px-KellsFol034rXRhoDet3.jpeg" decoding="async" width="190" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/KellsFol034rXRhoDet3.jpeg/285px-KellsFol034rXRhoDet3.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/KellsFol034rXRhoDet3.jpeg/380px-KellsFol034rXRhoDet3.jpeg 2x" data-file-width="487" data-file-height="496" /></a><figcaption>Intricate Celtic knotwork in the 1200-year-old <a href="/wiki/Book_of_Kells" title="Book of Kells">Book of Kells</a></figcaption></figure> <p>Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as <a href="/wiki/Khipu" class="mw-redirect" title="Khipu">recording information</a> and <a href="/wiki/Knot_tying" class="mw-redirect" title="Knot tying">tying</a> objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC (see <a href="/wiki/Chinese_knotting" title="Chinese knotting">Chinese knotting</a>). The <a href="/wiki/Endless_knot" title="Endless knot">endless knot</a> appears in <a href="/wiki/Tibetan_Buddhism" title="Tibetan Buddhism">Tibetan Buddhism</a>, while the <a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a> have made repeated appearances in different cultures, often representing strength in unity. The <a href="/wiki/Celtic_Christianity" title="Celtic Christianity">Celtic</a> monks who created the <a href="/wiki/Book_of_Kells" title="Book of Kells">Book of Kells</a> lavished entire pages with intricate <a href="/wiki/Celtic_knot" title="Celtic knot">Celtic knotwork</a>. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg/170px-Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg" decoding="async" width="170" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg/255px-Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg/340px-Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg 2x" data-file-width="2315" data-file-height="3017" /></a><figcaption>The first knot tabulator, <a href="/wiki/Peter_Guthrie_Tait" title="Peter Guthrie Tait">Peter Guthrie Tait</a></figcaption></figure> <p>A mathematical theory of knots was first developed in 1771 by <a href="/wiki/Alexandre-Th%C3%A9ophile_Vandermonde" title="Alexandre-Théophile Vandermonde">Alexandre-Théophile Vandermonde</a> who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>, who defined the <a href="/wiki/Linking_integral" class="mw-redirect" title="Linking integral">linking integral</a> (<a href="#CITEREFSilver2006">Silver 2006</a>). In the 1860s, <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">Lord Kelvin</a>'s <a href="/wiki/Vortex_theory_of_the_atom" title="Vortex theory of the atom">theory that atoms were knots in the aether</a> led to <a href="/wiki/Peter_Guthrie_Tait" title="Peter Guthrie Tait">Peter Guthrie Tait</a>'s creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, and what came to be known as the <a href="/wiki/Tait_conjectures" title="Tait conjectures">Tait conjectures</a>. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of <a href="/wiki/Topology" title="Topology">topology</a>. </p><p>These topologists in the early part of the 20th century—<a href="/wiki/Max_Dehn" title="Max Dehn">Max Dehn</a>, <a href="/wiki/James_Waddell_Alexander_II" title="James Waddell Alexander II">J. W. Alexander</a>, and others—studied knots from the point of view of the <a href="/wiki/Knot_group" title="Knot group">knot group</a> and invariants from <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> theory such as the <a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a>. This would be the main approach to knot theory until a series of breakthroughs transformed the subject. </p><p>In the late 1970s, <a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a> introduced <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> into the study of knots with the <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">hyperbolization theorem</a>. Many knots were shown to be <a href="/wiki/Hyperbolic_knot" class="mw-redirect" title="Hyperbolic knot">hyperbolic knots</a>, enabling the use of geometry in defining new, powerful <a href="/wiki/Knot_invariant" title="Knot invariant">knot invariants</a>. The discovery of the <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a> by <a href="/wiki/Vaughan_Jones" title="Vaughan Jones">Vaughan Jones</a> in 1984 (<a href="#CITEREFSossinsky2002">Sossinsky 2002</a>, pp. 71–89), and subsequent contributions from <a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a>, <a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Maxim Kontsevich</a>, and others, revealed deep connections between knot theory and mathematical methods in <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> and <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as <a href="/wiki/Quantum_group" title="Quantum group">quantum groups</a> and <a href="/wiki/Floer_homology" title="Floer homology">Floer homology</a>. </p><p>In the last several decades of the 20th century, scientists became interested in studying <a href="/wiki/Physical_knot_theory" title="Physical knot theory">physical knots</a> in order to understand knotting phenomena in <a href="/wiki/DNA" title="DNA">DNA</a> and other polymers. Knot theory can be used to determine if a molecule is <a href="/wiki/Chirality_(chemistry)" title="Chirality (chemistry)">chiral</a> (has a "handedness") or not (<a href="#CITEREFSimon1986">Simon 1986</a>). <a href="/wiki/Tangle_(mathematics)" title="Tangle (mathematics)">Tangles</a>, strings with both ends fixed in place, have been effectively used in studying the action of <a href="/wiki/Topoisomerase" title="Topoisomerase">topoisomerase</a> on DNA (<a href="#CITEREFFlapan2000">Flapan 2000</a>). Knot theory may be crucial in the construction of quantum computers, through the model of <a href="/wiki/Topological_quantum_computation" class="mw-redirect" title="Topological quantum computation">topological quantum computation</a> (<a href="#CITEREFCollins2006">Collins 2006</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Knot_equivalence">Knot equivalence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=2" title="Edit section: Knot equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:192px;max-width:192px"><div class="trow"><div class="tsingle" style="width:69px;max-width:69px"><div class="thumbimage" style="height:116px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Unknots.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Unknots.svg/67px-Unknots.svg.png" decoding="async" width="67" height="56" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Unknots.svg/101px-Unknots.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Unknots.svg/134px-Unknots.svg.png 2x" data-file-width="166" data-file-height="138" /></a></span></div></div><div class="tsingle" style="width:119px;max-width:119px"><div class="thumbimage" style="height:116px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Knot_Unfolding.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Knot_Unfolding.gif/117px-Knot_Unfolding.gif" decoding="async" width="117" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Knot_Unfolding.gif/176px-Knot_Unfolding.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Knot_Unfolding.gif/234px-Knot_Unfolding.gif 2x" data-file-width="700" data-file-height="700" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">On the left, the unknot, and a knot equivalent to it. It can be more difficult to determine whether complex knots, such as the one on the right, are equivalent to the unknot.</div></div></div></div> <p>A knot is created by beginning with a one-<a href="/wiki/Dimension" title="Dimension">dimensional</a> line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop (<a href="#CITEREFAdams2004">Adams 2004</a>) (<a href="#CITEREFSossinsky2002">Sossinsky 2002</a>). Simply, we can say a knot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is a "simple closed curve" (see <a href="/wiki/Curve" title="Curve">Curve</a>) — that is: a "nearly" <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a> and <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\colon [0,1]\to \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\colon [0,1]\to \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3527c328346ff511b17bd2fe2ae5f3504df3d2e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.099ex; height:3.176ex;" alt="{\displaystyle K\colon [0,1]\to \mathbb {R} ^{3}}"></span>, with the only "non-injectivity" being <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(0)=K(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(0)=K(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b777ceb922b136715855af6364a3c8e71fc399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.174ex; height:2.843ex;" alt="{\displaystyle K(0)=K(1)}"></span>. Topologists consider knots and other entanglements such as <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">links</a> and <a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">braids</a> to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot. </p><p>The idea of <b>knot equivalence</b> is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A formal mathematical definition is that two knots <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1},K_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1},K_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e1ca4f18aed738f059171d774a63a07835d325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.088ex; height:2.509ex;" alt="{\displaystyle K_{1},K_{2}}"></span> are equivalent if there is an <a href="/wiki/Orientation-preserving" class="mw-redirect" title="Orientation-preserving">orientation-preserving</a> <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c0019c7ed674928b50ef302901844595448582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.452ex; height:2.676ex;" alt="{\displaystyle h\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(K_{1})=K_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(K_{1})=K_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9816f3eae5285f249b1725b04093a81ade864522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.301ex; height:2.843ex;" alt="{\displaystyle h(K_{1})=K_{2}}"></span>. </p><p>What this definition of knot equivalence means is that two knots are equivalent when there is a continuous family of homeomorphisms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{h_{t}:\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}\ \mathrm {for} \ 0\leq t\leq 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{h_{t}:\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}\ \mathrm {for} \ 0\leq t\leq 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178d69636ac401950e3f90e605e93578ec894d22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.968ex; height:3.176ex;" alt="{\displaystyle \{h_{t}:\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}\ \mathrm {for} \ 0\leq t\leq 1\}}"></span> of space onto itself, such that the last one of them carries the first knot onto the second knot. (In detail: Two knots <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e1b324cf5b68f2729a8634ff76e396b634b75d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{2}}"></span> are <b>equivalent</b> if there exists a continuous mapping <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:\mathbb {R} ^{3}\times [0,1]\rightarrow \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H:\mathbb {R} ^{3}\times [0,1]\rightarrow \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc228092f3ea7de7bbd32a579a7498ea08e9fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.572ex; height:3.176ex;" alt="{\displaystyle H:\mathbb {R} ^{3}\times [0,1]\rightarrow \mathbb {R} ^{3}}"></span> such that a) for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a5c18739ff04858eecc8fec2f53912c348e0e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:2.843ex;" alt="{\displaystyle t\in [0,1]}"></span> the mapping taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04753232bfa03dc57d5dbf46f4f122e62f99a324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.903ex; height:2.676ex;" alt="{\displaystyle x\in \mathbb {R} ^{3}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,t)\in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,t)\in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acbea4adbce617fd4bf5ff7f417f6cc361a9dc35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.649ex; height:3.176ex;" alt="{\displaystyle H(x,t)\in \mathbb {R} ^{3}}"></span> is a homeomorphism of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> onto itself; b) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x,0)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x,0)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9181aff6335c0d9d177e25026ba437e348e8aac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.827ex; height:2.843ex;" alt="{\displaystyle H(x,0)=x}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04753232bfa03dc57d5dbf46f4f122e62f99a324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.903ex; height:2.676ex;" alt="{\displaystyle x\in \mathbb {R} ^{3}}"></span>; and c) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(K_{1},1)=K_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(K_{1},1)=K_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d81fda6cc35678b8855cd3d6640e3c7d67f1766c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.222ex; height:2.843ex;" alt="{\displaystyle H(K_{1},1)=K_{2}}"></span>. Such a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is known as an <a href="/wiki/Ambient_isotopy" title="Ambient isotopy">ambient isotopy</a>.) </p><p>These two notions of knot equivalence agree exactly about which knots are equivalent: Two knots that are equivalent under the orientation-preserving homeomorphism definition are also equivalent under the ambient isotopy definition, because any orientation-preserving homeomorphisms of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> to itself is the final stage of an ambient isotopy starting from the identity. Conversely, two knots equivalent under the ambient isotopy definition are also equivalent under the orientation-preserving homeomorphism definition, because the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970dea4a5f5ec5355c4cdd62f6396fbc8b1baaa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=1}"></span> (final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other. </p><p>The basic problem of knot theory, the <b>recognition problem</b>, is determining the equivalence of two knots. <a href="/wiki/Algorithm" title="Algorithm">Algorithms</a> exist to solve this problem, with the first given by <a href="/wiki/Wolfgang_Haken" title="Wolfgang Haken">Wolfgang Haken</a> in the late 1960s (<a href="#CITEREFHass1998">Hass 1998</a>). Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is (<a href="#CITEREFHass1998">Hass 1998</a>). The special case of recognizing the <a href="/wiki/Unknot" title="Unknot">unknot</a>, called the <a href="/wiki/Unknotting_problem" title="Unknotting problem">unknotting problem</a>, is of particular interest (<a href="#CITEREFHoste2005">Hoste 2005</a>). In February 2021 <a href="/wiki/Marc_Lackenby" title="Marc Lackenby">Marc Lackenby</a> announced a new unknot recognition algorithm that runs in <a href="/wiki/Time_complexity" title="Time complexity">quasi-polynomial time</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Knot_diagrams">Knot diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=3" title="Edit section: Knot diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tenfold_Knottiness,_plate_IX.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tenfold_Knottiness%2C_plate_IX.png/220px-Tenfold_Knottiness%2C_plate_IX.png" decoding="async" width="220" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tenfold_Knottiness%2C_plate_IX.png/330px-Tenfold_Knottiness%2C_plate_IX.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tenfold_Knottiness%2C_plate_IX.png/440px-Tenfold_Knottiness%2C_plate_IX.png 2x" data-file-width="1918" data-file-height="2500" /></a><figcaption>Tenfold Knottiness, plate IX, from <a href="/wiki/Peter_Guthrie_Tait" title="Peter Guthrie Tait">Peter Guthrie Tait</a>'s article "On Knots", 1884</figcaption></figure> <p>A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is <a href="/wiki/Injective_function" title="Injective function">one-to-one</a> except at the double points, called <i>crossings</i>, where the "shadow" of the knot crosses itself once transversely (<a href="#CITEREFRolfsen1976">Rolfsen 1976</a>). At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath. The resulting diagram is an <a href="/wiki/Immersed_plane_curve" class="mw-redirect" title="Immersed plane curve">immersed plane curve</a> with the additional data of which strand is over and which is under at each crossing. (These diagrams are called <b>knot diagrams</b> when they represent a <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot</a> and <b>link diagrams</b> when they represent a <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">link</a>.) Analogously, knotted surfaces in 4-space can be related to <a href="/wiki/Immersed_surface" class="mw-redirect" title="Immersed surface">immersed surfaces</a> in 3-space. </p><p>A <b>reduced diagram</b> is a knot diagram in which there are no <b>reducible crossings</b> (also <b>nugatory</b> or <b>removable crossings</b>), or in which all of the reducible crossings have been removed.<sup id="cite_ref-FOOTNOTEWeisstein2013_3-0" class="reference"><a href="#cite_note-FOOTNOTEWeisstein2013-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEWeisstein2013a_4-0" class="reference"><a href="#cite_note-FOOTNOTEWeisstein2013a-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A <a href="/wiki/Petal_projection" title="Petal projection">petal projection</a> is a type of projection in which, instead of forming double points, all strands of the knot meet at a single crossing point, connected to it by loops forming non-nested "petals".<sup id="cite_ref-FOOTNOTEAdamsCrawfordDeMeoLandry2015_5-0" class="reference"><a href="#cite_note-FOOTNOTEAdamsCrawfordDeMeoLandry2015-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Reidemeister_moves">Reidemeister moves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=4" title="Edit section: Reidemeister moves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Reidemeister_move" title="Reidemeister move">Reidemeister move</a></div> <p>In 1927, working with this diagrammatic form of knots, <a href="/wiki/James_Waddell_Alexander_II" title="James Waddell Alexander II">J. W. Alexander</a> and <a href="/w/index.php?title=Garland_Baird_Briggs&action=edit&redlink=1" class="new" title="Garland Baird Briggs (page does not exist)">Garland Baird Briggs</a>, and independently <a href="/wiki/Kurt_Reidemeister" title="Kurt Reidemeister">Kurt Reidemeister</a>, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the <i>Reidemeister moves</i>, are: </p> <div><ol style="list-style-type:upper-Roman"><li>Twist and untwist in either direction.</li><li>Move one strand completely over another.</li><li>Move a strand completely over or under a crossing.</li></ol></div> <table align="center" style="text-align:center"> <caption><b>Reidemeister moves</b> </caption> <tbody><tr style="padding:1em"> <td><span typeof="mw:File"><a href="/wiki/File:Reidemeister_move_1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Reidemeister_move_1.png/130px-Reidemeister_move_1.png" decoding="async" width="130" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Reidemeister_move_1.png/195px-Reidemeister_move_1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b1/Reidemeister_move_1.png 2x" data-file-width="260" data-file-height="309" /></a></span> <span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Frame_left.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/67/Frame_left.png" decoding="async" width="5" height="133" class="mw-file-element" data-file-width="5" data-file-height="133" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:Reidemeister_move_2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Reidemeister_move_2.png/210px-Reidemeister_move_2.png" decoding="async" width="210" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Reidemeister_move_2.png/315px-Reidemeister_move_2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/60/Reidemeister_move_2.png 2x" data-file-width="420" data-file-height="322" /></a></span> </td></tr> <tr> <th>Type I</th> <th>Type II </th></tr> <tr style="padding:1em"> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Reidemeister_move_3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Reidemeister_move_3.png/360px-Reidemeister_move_3.png" decoding="async" width="360" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Reidemeister_move_3.png/540px-Reidemeister_move_3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Reidemeister_move_3.png/720px-Reidemeister_move_3.png 2x" data-file-width="725" data-file-height="289" /></a></span> </td></tr> <tr> <th colspan="2">Type III </th></tr></tbody></table> <p>The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another. The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point. A close inspection will show that complicated events can be eliminated, leaving only the simplest events: (1) a "kink" forming or being straightened out; (2) two strands becoming tangent at a point and passing through; and (3) three strands crossing at a point. These are precisely the Reidemeister moves (<a href="#CITEREFSossinsky2002">Sossinsky 2002</a>, ch. 3) (<a href="#CITEREFLickorish1997">Lickorish 1997</a>, ch. 1). </p> <div class="mw-heading mw-heading2"><h2 id="Knot_invariants">Knot invariants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=5" title="Edit section: Knot invariants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Figure_eight_knot_complement.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Figure_eight_knot_complement.jpg/280px-Figure_eight_knot_complement.jpg" decoding="async" width="280" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Figure_eight_knot_complement.jpg/420px-Figure_eight_knot_complement.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Figure_eight_knot_complement.jpg/560px-Figure_eight_knot_complement.jpg 2x" data-file-width="3530" data-file-height="2647" /></a><figcaption> A 3D print depicting the complement of the figure eight knot<br />by François Guéritaud, Saul Schleimer, and <a href="/wiki/Henry_Segerman" title="Henry Segerman">Henry Segerman</a></figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Knot_invariant" title="Knot invariant">Knot invariant</a></div> <p>A knot invariant is a "quantity" that is the same for equivalent knots (<a href="#CITEREFAdams2004">Adams 2004</a>) (<a href="#CITEREFLickorish1997">Lickorish 1997</a>) (<a href="#CITEREFRolfsen1976">Rolfsen 1976</a>). For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots. An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is <a href="/wiki/Tricolorability" title="Tricolorability">tricolorability</a>. </p><p>"Classical" knot invariants include the <a href="/wiki/Knot_group" title="Knot group">knot group</a>, which is the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of the <a href="/wiki/Knot_complement" title="Knot complement">knot complement</a>, and the <a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a>, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement (<a href="#CITEREFLickorish1997">Lickorish 1997</a>)(<a href="#CITEREFRolfsen1976">Rolfsen 1976</a>). In the late 20th century, invariants such as "quantum" knot polynomials, <a href="/wiki/Vassiliev_invariant" class="mw-redirect" title="Vassiliev invariant">Vassiliev invariants</a> and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory. </p> <div class="mw-heading mw-heading3"><h3 id="Knot_polynomials">Knot polynomials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=6" title="Edit section: Knot polynomials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Knot_polynomial" title="Knot polynomial">Knot polynomial</a></div> <p>A knot polynomial is a <a href="/wiki/Knot_invariant" title="Knot invariant">knot invariant</a> that is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>. Well-known examples include the <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a>, the <a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a>, and the <a href="/wiki/Kauffman_polynomial" title="Kauffman polynomial">Kauffman polynomial</a>. A variant of the Alexander polynomial, the <a href="/wiki/Alexander%E2%80%93Conway_polynomial" class="mw-redirect" title="Alexander–Conway polynomial">Alexander–Conway polynomial</a>, is a polynomial in the variable <i>z</i> with <a href="/wiki/Integer" title="Integer">integer</a> coefficients (<a href="#CITEREFLickorish1997">Lickorish 1997</a>). </p><p>The Alexander–Conway polynomial is actually defined in terms of <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">links</a>, which consist of one or more knots entangled with each other. The concepts explained above for knots, e.g. diagrams and Reidemeister moves, also hold for links. </p><p> Consider an oriented link diagram, <i>i.e.</i> one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{+},L_{-},L_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{+},L_{-},L_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c068537cef53c832e1388400f99b3a33892af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.892ex; height:2.509ex;" alt="{\displaystyle L_{+},L_{-},L_{0}}"></span> be the oriented link diagrams resulting from changing the diagram as indicated in the figure: </p><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Skein_(HOMFLY).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Skein_%28HOMFLY%29.svg/200px-Skein_%28HOMFLY%29.svg.png" decoding="async" width="200" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Skein_%28HOMFLY%29.svg/300px-Skein_%28HOMFLY%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Skein_%28HOMFLY%29.svg/400px-Skein_%28HOMFLY%29.svg.png 2x" data-file-width="300" data-file-height="160" /></a><figcaption></figcaption></figure> <p>The original diagram might be either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0400c11cecda7bb058ba191989776ed7b60d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.094ex; height:2.509ex;" alt="{\displaystyle L_{+}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e31496c2dfd353cf10c89b0d86a82831c71abf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.094ex; height:2.509ex;" alt="{\displaystyle L_{-}}"></span>, depending on the chosen crossing's configuration. Then the Alexander–Conway polynomial, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f93d5873532147d908311b46a43b70422d3503b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.664ex; height:2.843ex;" alt="{\displaystyle C(z)}"></span>, is recursively defined according to the rules: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(O)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(O)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e504b761be316c60a186b669dfc0af65e46f334b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.61ex; height:2.843ex;" alt="{\displaystyle C(O)=1}"></span> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> is any diagram of the <a href="/wiki/Unknot" title="Unknot">unknot</a>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(L_{+})=C(L_{-})+zC(L_{0}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(L_{+})=C(L_{-})+zC(L_{0}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ecfa29deebdf7543f498bf9e07b2d63b2892a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.225ex; height:2.843ex;" alt="{\displaystyle C(L_{+})=C(L_{-})+zC(L_{0}).}"></span></li></ul> <p>The second rule is what is often referred to as a <a href="/wiki/Skein_relation" title="Skein relation">skein relation</a>. To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way. </p><p>The following is an example of a typical computation using a skein relation. It computes the Alexander–Conway polynomial of the <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knot</a>. The yellow patches indicate where the relation is applied. </p> <dl><dd><i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-trefoil-plus-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/db/Skein-relation-trefoil-plus-sm.png" decoding="async" width="47" height="48" class="mw-file-element" data-file-width="47" data-file-height="48" /></a></span>) = <i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-trefoil-minus-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/99/Skein-relation-trefoil-minus-sm.png" decoding="async" width="47" height="48" class="mw-file-element" data-file-width="47" data-file-height="48" /></a></span>) + <i>z</i> <i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-trefoil-zero-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Skein-relation-trefoil-zero-sm.png" decoding="async" width="47" height="48" class="mw-file-element" data-file-width="47" data-file-height="48" /></a></span>)</dd></dl> <p>gives the unknot and the <a href="/wiki/Hopf_link" title="Hopf link">Hopf link</a>. Applying the relation to the Hopf link where indicated, </p> <dl><dd><i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-link22-plus-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/ba/Skein-relation-link22-plus-sm.png" decoding="async" width="44" height="48" class="mw-file-element" data-file-width="44" data-file-height="48" /></a></span>) = <i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-link22-minus-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/1a/Skein-relation-link22-minus-sm.png" decoding="async" width="44" height="48" class="mw-file-element" data-file-width="44" data-file-height="48" /></a></span>) + <i>z</i> <i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-link22-zero-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a9/Skein-relation-link22-zero-sm.png" decoding="async" width="44" height="48" class="mw-file-element" data-file-width="44" data-file-height="48" /></a></span>)</dd></dl> <p>gives a link deformable to one with 0 crossings (it is actually the <a href="/wiki/Unlink" title="Unlink">unlink</a> of two components) and an unknot. The unlink takes a bit of sneakiness: </p> <dl><dd><i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-link20-plus-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/1e/Skein-relation-link20-plus-sm.png" decoding="async" width="48" height="48" class="mw-file-element" data-file-width="48" data-file-height="48" /></a></span>) = <i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-link20-minus-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/73/Skein-relation-link20-minus-sm.png" decoding="async" width="48" height="48" class="mw-file-element" data-file-width="48" data-file-height="48" /></a></span>) + <i>z</i> <i>C</i>(<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Skein-relation-link20-zero-sm.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/ab/Skein-relation-link20-zero-sm.png" decoding="async" width="48" height="48" class="mw-file-element" data-file-width="48" data-file-height="48" /></a></span>)</dd></dl> <p>which implies that <i>C</i>(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal. </p><p>Putting all this together will show: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\mathrm {trefoil} )=1+z(0+z)=1+z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathrm {trefoil} )=1+z(0+z)=1+z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbadee567dd1017599001d6546a30a3946794c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.081ex; height:3.176ex;" alt="{\displaystyle C(\mathrm {trefoil} )=1+z(0+z)=1+z^{2}}"></span></dd></dl> <p>Since the Alexander–Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted". </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 115px"> <div class="thumb" style="width: 110px; height: 110px;"><span typeof="mw:File"><a href="/wiki/File:Trefoil_knot_left.svg" class="mw-file-description" title="The left-handed trefoil knot."><img alt="The left-handed trefoil knot." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Trefoil_knot_left.svg/80px-Trefoil_knot_left.svg.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Trefoil_knot_left.svg/120px-Trefoil_knot_left.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Trefoil_knot_left.svg/160px-Trefoil_knot_left.svg.png 2x" data-file-width="250" data-file-height="250" /></a></span></div> <div class="gallerytext">The left-handed trefoil knot.</div> </li> <li class="gallerybox" style="width: 115px"> <div class="thumb" style="width: 110px; height: 110px;"><span typeof="mw:File"><a href="/wiki/File:TrefoilKnot_01.svg" class="mw-file-description" title="The right-handed trefoil knot."><img alt="The right-handed trefoil knot." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/80px-TrefoilKnot_01.svg.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/120px-TrefoilKnot_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/160px-TrefoilKnot_01.svg.png 2x" data-file-width="250" data-file-height="250" /></a></span></div> <div class="gallerytext">The right-handed trefoil knot.</div> </li> </ul> <p>Actually, there are two trefoil knots, called the right and left-handed trefoils, which are <a href="/wiki/Chiral_knot" title="Chiral knot">mirror images</a> of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). These are not equivalent to each other, meaning that they are not amphichiral. This was shown by <a href="/wiki/Max_Dehn" title="Max Dehn">Max Dehn</a>, before the invention of knot polynomials, using group theoretical methods (<a href="#CITEREFDehn1914">Dehn 1914</a>). But the Alexander–Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The <i>Jones</i> polynomial can in fact distinguish between the left- and right-handed trefoil knots (<a href="#CITEREFLickorish1997">Lickorish 1997</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Hyperbolic_invariants">Hyperbolic invariants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=7" title="Edit section: Hyperbolic invariants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a> proved many knots are <a href="/wiki/Hyperbolic_knot" class="mw-redirect" title="Hyperbolic knot">hyperbolic knots</a>, meaning that the <a href="/wiki/Knot_complement" title="Knot complement">knot complement</a> (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant (<a href="#CITEREFAdams2004">Adams 2004</a>). </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:312px;max-width:312px"><div class="trow"><div class="tsingle" style="width:135px;max-width:135px"><div class="thumbimage" style="height:127px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:BorromeanRings.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/BorromeanRings.svg/133px-BorromeanRings.svg.png" decoding="async" width="133" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/BorromeanRings.svg/200px-BorromeanRings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/BorromeanRings.svg/266px-BorromeanRings.svg.png 2x" data-file-width="626" data-file-height="600" /></a></span></div><div class="thumbcaption">The <a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a> are a link with the property that removing one ring unlinks the others.</div></div><div class="tsingle" style="width:173px;max-width:173px"><div class="thumbimage" style="height:127px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:SnapPea-horocusp_view.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/SnapPea-horocusp_view.png/171px-SnapPea-horocusp_view.png" decoding="async" width="171" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/SnapPea-horocusp_view.png/257px-SnapPea-horocusp_view.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/SnapPea-horocusp_view.png/342px-SnapPea-horocusp_view.png 2x" data-file-width="560" data-file-height="416" /></a></span></div><div class="thumbcaption"><a href="/wiki/SnapPea" title="SnapPea">SnapPea</a>'s cusp view: the <a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a> complement from the perspective of an inhabitant living near the red component.</div></div></div></div></div> <p>Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the <a href="/wiki/Geodesic" title="Geodesic">geodesics</a> of the geometry. An example is provided by the picture of the complement of the <a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a>. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of <a href="/wiki/Horoball" class="mw-redirect" title="Horoball">horoball</a> neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely. </p><p>This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task (<a href="#CITEREFAdamsHildebrandWeeks1991">Adams, Hildebrand & Weeks 1991</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Higher_dimensions">Higher dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=8" title="Edit section: Higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A knot in three dimensions can be untied when placed in four-dimensional space. This is done by changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle. </p><p>In fact, in four dimensions, any non-intersecting closed loop of one-dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain. </p><p>Four-dimensional space occurs in classical knot theory, however, and an important topic is the study of <a href="/wiki/Slice_knot" title="Slice knot">slice knots</a> and <a href="/wiki/Ribbon_knot" title="Ribbon knot">ribbon knots</a>. A notorious open problem asks whether every slice knot is also ribbon. </p> <div class="mw-heading mw-heading3"><h3 id="Knotting_spheres_of_higher_dimension">Knotting spheres of higher dimension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=9" title="Edit section: Knotting spheres of higher dimension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a <a href="/wiki/2-sphere" class="mw-redirect" title="2-sphere">two-dimensional sphere</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/518268b00344a37811c08a236412bffaa68f75d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.347ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{2}}"></span>) embedded in 4-dimensional Euclidean space (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span>). Such an embedding is knotted if there is no homeomorphism of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span> onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere. <a href="/w/index.php?title=Suspended_knot&action=edit&redlink=1" class="new" title="Suspended knot (page does not exist)">Suspended knots</a> and <a href="/w/index.php?title=Spun_knot&action=edit&redlink=1" class="new" title="Spun knot (page does not exist)">spun knots</a> are two typical families of such 2-sphere knots. </p><p>The mathematical technique called "general position" implies that for a given <i>n</i>-sphere in <i>m</i>-dimensional Euclidean space, if <i>m</i> is large enough (depending on <i>n</i>), the sphere should be unknotted. In general, <a href="/wiki/Piecewise_linear_manifold" title="Piecewise linear manifold">piecewise-linear</a> <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-spheres</a> form knots only in (<i>n</i> + 2)-dimensional space (<a href="#CITEREFZeeman1963">Zeeman 1963</a>), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4k-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4k-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7eae7a52f5a332ce92a85d0aea500b95512dfe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.186ex; height:2.843ex;" alt="{\displaystyle (4k-1)}"></span>-spheres in 6<i>k</i>-dimensional space; e.g., there is a smoothly knotted 3-sphere in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547484e275658bac48b1ad8f5407446612d4a65c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{6}}"></span> (<a href="#CITEREFHaefliger1962">Haefliger 1962</a>) (<a href="#CITEREFLevine1965">Levine 1965</a>). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth <i>k</i>-sphere embedded in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n-3k-3>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mi>k</mi> <mo>−</mo> <mn>3</mn> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n-3k-3>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f4bd67175fe6d6b322b56055f7a3c19a0ea599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.035ex; height:2.343ex;" alt="{\displaystyle 2n-3k-3>0}"></span> is unknotted. The notion of a knot has further generalisations in mathematics, see: <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">Knot (mathematics)</a>, <a href="/wiki/Whitney_embedding_theorem#Isotopy_versions" title="Whitney embedding theorem">isotopy classification of embeddings</a>. </p><p>Every knot in the <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-sphere</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c00c9d3635f5230e6ac11902a50f2323794ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.343ex;" alt="{\displaystyle \mathbb {S} ^{n}}"></span> is the link of a <a href="/wiki/Real_algebraic_set" class="mw-redirect" title="Real algebraic set">real-algebraic set</a> with isolated singularity in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccea3976e1f8a1bb853c8ca00e52d518a3a4fe07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.997ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n+1}}"></span> (<a href="#CITEREFAkbulutKing1981">Akbulut & King 1981</a>). </p><p>An <i>n</i>-knot is a single <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c00c9d3635f5230e6ac11902a50f2323794ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.343ex;" alt="{\displaystyle \mathbb {S} ^{n}}"></span> embedded in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}"></span>. An <i>n</i>-link consists of <i>k</i>-copies of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c00c9d3635f5230e6ac11902a50f2323794ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.343ex;" alt="{\displaystyle \mathbb {S} ^{n}}"></span> embedded in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}"></span>, where <i>k</i> is a <a href="/wiki/Natural_number" title="Natural number">natural number</a>. Both the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=n+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=n+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd7a87626645eebe52faa6a0b2b019ec7279465" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.536ex; height:2.343ex;" alt="{\displaystyle m=n+2}"></span> and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>n+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>n+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a777e81e5204aeb3379fdfa516e7dff5bdf02b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.536ex; height:2.343ex;" alt="{\displaystyle m>n+2}"></span> cases are well studied, and so is the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"></span> case.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Adding_knots">Adding knots</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=10" title="Edit section: Adding knots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Knot sum</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sum_of_knots3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Sum_of_knots3.svg/220px-Sum_of_knots3.svg.png" decoding="async" width="220" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Sum_of_knots3.svg/330px-Sum_of_knots3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Sum_of_knots3.svg/440px-Sum_of_knots3.svg.png 2x" data-file-width="300" data-file-height="120" /></a><figcaption>Adding two knots</figcaption></figure> <p>Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the <i>knot sum</i>, or sometimes the <i>connected sum</i> or <i>composition</i> of two knots. This can be formally defined as follows (<a href="#CITEREFAdams2004">Adams 2004</a>): consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is a sum of the original knots. Depending on how this is done, two different knots (but no more) may result. This ambiguity in the sum can be eliminated regarding the knots as <i>oriented</i>, i.e. having a preferred direction of travel along the knot, and requiring the arcs of the knots in the sum are oriented consistently with the oriented boundary of the rectangle. </p><p>The knot sum of oriented knots is <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> and <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>. A <a href="/wiki/Prime_knot" title="Prime knot">knot is <i>prime</i></a> if it is non-trivial and cannot be written as the knot sum of two non-trivial knots. A knot that can be written as such a sum is <i>composite</i>. There is a prime decomposition for knots, analogous to <a href="/wiki/Prime_number" title="Prime number">prime</a> and composite numbers (<a href="#CITEREFSchubert1949">Schubert 1949</a>). For oriented knots, this decomposition is also unique. Higher-dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers <i>smooth</i> knots in codimension at least 3. </p><p>Knots can also be constructed using the <a href="/wiki/Circuit_topology" title="Circuit topology">circuit topology</a> approach. This is done by combining basic units called soft contacts using five operations (Parallel, Series, Cross, Concerted, and Sub).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The approach is applicable to open chains as well and can also be extended to include the so-called hard contacts. </p> <div class="mw-heading mw-heading2"><h2 id="Tabulating_knots">Tabulating knots</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=11" title="Edit section: Tabulating knots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_prime_knots" title="List of prime knots">List of prime knots</a> and <a href="/wiki/Knot_tabulation" title="Knot tabulation">Knot tabulation</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Knot_table.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Knot_table.svg/330px-Knot_table.svg.png" decoding="async" width="330" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Knot_table.svg/495px-Knot_table.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Knot_table.svg/660px-Knot_table.svg.png 2x" data-file-width="470" data-file-height="350" /></a><figcaption>A table of prime knots up to seven crossings. The knots are labeled with Alexander–Briggs notation</figcaption></figure> <p>Traditionally, knots have been catalogued in terms of <a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">crossing number</a>. Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) (<a href="#CITEREFHosteThistlethwaiteWeeks1998">Hoste, Thistlethwaite & Weeks 1998</a>). The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult (<a href="#CITEREFHoste2005">Hoste 2005</a>, p. 20). Tabulation efforts have succeeded in enumerating over 6 billion knots and links (<a href="#CITEREFHoste2005">Hoste 2005</a>, p. 28). The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, <span class="nowrap"><span data-sort-value="7004469720000000000♠"></span>46<span style="margin-left:.25em;">972</span></span>, <span class="nowrap"><span data-sort-value="7005253293000000000♠"></span>253<span style="margin-left:.25em;">293</span></span>, <span class="nowrap"><span data-sort-value="7006138870500000000♠"></span>1<span style="margin-left:.25em;">388</span><span style="margin-left:.25em;">705</span></span>... (sequence <span class="nowrap external"><a href="//oeis.org/A002863" class="extiw" title="oeis:A002863">A002863</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing (<a href="#CITEREFAdams2004">Adams 2004</a>). </p><p>The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the <a href="/wiki/Dowker_notation" class="mw-redirect" title="Dowker notation">Dowker notation</a>. Different notations have been invented for knots which allow more efficient tabulation (<a href="#CITEREFHoste2005">Hoste 2005</a>). </p><p>The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings (<a href="#CITEREFHosteThistlethwaiteWeeks1998">Hoste, Thistlethwaite & Weeks 1998</a>). The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander–Briggs and Reidemeister in the late 1920s. </p><p>The first major verification of this work was done in the 1960s by <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Horton Conway</a>, who not only developed a new notation but also the <a href="/wiki/Alexander%E2%80%93Conway_polynomial" class="mw-redirect" title="Alexander–Conway polynomial">Alexander–Conway polynomial</a> (<a href="#CITEREFConway1970">Conway 1970</a>) (<a href="#CITEREFDollHoste1991">Doll & Hoste 1991</a>). This verified the list of knots of at most 11 crossings and a new list of links up to 10 crossings. Conway found a number of omissions but only one duplication in the Tait–Little tables; however he missed the duplicates called the <a href="/wiki/Perko_pair" title="Perko pair">Perko pair</a>, which would only be noticed in 1974 by <a href="/wiki/Kenneth_Perko" class="mw-redirect" title="Kenneth Perko">Kenneth Perko</a> (<a href="#CITEREFPerko1974">Perko 1974</a>). This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work. Conway's 1970 paper on knot theory also contains a typographical duplication on its non-alternating 11-crossing knots page and omits 4 examples — 2 previously listed in D. Lombardero's 1968 Princeton senior thesis and 2 more subsequently discovered by <a href="/w/index.php?title=Alain_Caudron&action=edit&redlink=1" class="new" title="Alain Caudron (page does not exist)">Alain Caudron</a>. [see Perko (1982), Primality of certain knots, Topology Proceedings] Less famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20. [See Perko (2016), Historical highlights of non-cyclic knot theory, J. Knot Theory Ramifications]. </p><p>In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings (<a href="#CITEREFHosteThistlethwaiteWeeks1998">Hoste, Thistlethwaite & Weeks 1998</a>). In 2003 Rankin, Flint, and Schermann, tabulated the <a href="/wiki/Alternating_knot" title="Alternating knot">alternating knots</a> through 22 crossings (<a href="#CITEREFHoste2005">Hoste 2005</a>). In 2020 Burton tabulated all <a href="/wiki/Prime_knot" title="Prime knot">prime knots</a> with up to 19 crossings (<a href="#CITEREFBurton2020">Burton 2020</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Alexander–Briggs_notation"><span id="Alexander.E2.80.93Briggs_notation"></span>Alexander–Briggs notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=12" title="Edit section: Alexander–Briggs notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This is the most traditional notation, due to the 1927 paper of <a href="/wiki/James_Waddell_Alexander_II" title="James Waddell Alexander II">James W. Alexander</a> and <a href="/w/index.php?title=Garland_Baird_Briggs&action=edit&redlink=1" class="new" title="Garland Baird Briggs (page does not exist)">Garland B. Briggs</a> and later extended by <a href="/wiki/Dale_Rolfsen" class="mw-redirect" title="Dale Rolfsen">Dale Rolfsen</a> in his knot table (see image above and <a href="/wiki/List_of_prime_knots" title="List of prime knots">List of prime knots</a>). The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance (though in each number of crossings the <a href="/wiki/Twist_knot" title="Twist knot">twist knot</a> comes after the <a href="/wiki/Torus_knot" title="Torus knot">torus knot</a>). Links are written by the crossing number with a superscript to denote the number of components and a subscript to denote its order within the links with the same number of components and crossings. Thus the trefoil knot is notated 3<sub>1</sub> and the Hopf link is 2<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>. Alexander–Briggs names in the range 10<sub>162</sub> to 10<sub>166</sub> are ambiguous, due to the discovery of the <a href="/wiki/Perko_pair" title="Perko pair">Perko pair</a> in <a href="/wiki/Charles_Newton_Little" title="Charles Newton Little">Charles Newton Little</a>'s original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Dowker–Thistlethwaite_notation"><span id="Dowker.E2.80.93Thistlethwaite_notation"></span>Dowker–Thistlethwaite notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=13" title="Edit section: Dowker–Thistlethwaite notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dowker%E2%80%93Thistlethwaite_notation" title="Dowker–Thistlethwaite notation">Dowker–Thistlethwaite notation</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dowker-notation-example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Dowker-notation-example.svg/190px-Dowker-notation-example.svg.png" decoding="async" width="190" height="306" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Dowker-notation-example.svg/285px-Dowker-notation-example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Dowker-notation-example.svg/380px-Dowker-notation-example.svg.png 2x" data-file-width="248" data-file-height="400" /></a><figcaption>A knot diagram with crossings labelled for a Dowker sequence</figcaption></figure> <p>The <a href="/wiki/Dowker%E2%80%93Thistlethwaite_notation" title="Dowker–Thistlethwaite notation">Dowker–Thistlethwaite notation</a>, also called the Dowker notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in this figure the knot diagram has crossings labelled with the pairs (1,6) (3,−12) (5,2) (7,8) (9,−4) and (11,−10). The Dowker–Thistlethwaite notation for this labelling is the sequence: 6, −12, 2, 8, −4, −10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker–Thistlethwaite notation. </p> <div class="mw-heading mw-heading3"><h3 id="Conway_notation">Conway notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=14" title="Edit section: Conway notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation (knot theory)</a></div> <p>The <a href="/wiki/Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation</a> for knots and links, named after <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Horton Conway</a>, is based on the theory of <a href="/wiki/Tangle_(mathematics)" title="Tangle (mathematics)">tangles</a> (<a href="#CITEREFConway1970">Conway 1970</a>). The advantage of this notation is that it reflects some properties of the knot or link. </p><p>The notation describes how to construct a particular link diagram of the link. Start with a <i>basic polyhedron</i>, a 4-valent connected planar graph with no <a href="/wiki/Digon" title="Digon">digon</a> regions. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedra. For example, 10** denotes the second 10-vertex polyhedron on Conway's list. </p><p>Each vertex then has an <a href="/wiki/Algebraic_tangle" class="mw-redirect" title="Algebraic tangle">algebraic tangle</a> substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). Each such tangle has a notation consisting of numbers and + or − signs. </p><p>An example is 1*2 −3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 −3 2 is a sequence describing the continued fraction associated to a <a href="/wiki/Rational_tangle" class="mw-redirect" title="Rational tangle">rational tangle</a>. One inserts this tangle at the vertex of the basic polyhedron 1*. </p><p>A more complicated example is 8*3.1.2 0.1.1.1.1.1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle. </p><p>Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where the ones are omitted and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted. </p><p>Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available. </p> <div class="mw-heading mw-heading3"><h3 id="Gauss_code">Gauss code</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=15" title="Edit section: Gauss code"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gauss_code" class="mw-redirect" title="Gauss code">Gauss code</a></div> <p><a href="/wiki/Gauss_code" class="mw-redirect" title="Gauss code">Gauss code</a>, similar to the Dowker–Thistlethwaite notation, represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number. For example, the trefoil knot in Gauss code can be given as: 1,−2,3,−1,2,−3 </p><p>Gauss code is limited in its ability to identify knots. This problem is partially addressed with by the <a href="/wiki/Extended_Gauss_code" class="mw-redirect" title="Extended Gauss code">extended Gauss code</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Arithmetic_rope&action=edit&redlink=1" class="new" title="Arithmetic rope (page does not exist)">Arithmetic rope</a></li> <li><a href="/wiki/Circuit_topology" title="Circuit topology">Circuit topology</a></li> <li><a href="/wiki/Lamp_cord_trick" title="Lamp cord trick">Lamp cord trick</a></li> <li><a href="/wiki/Contact_geometry#Legendrian_submanifolds_and_knots" title="Contact geometry">Legendrian submanifolds and knots</a></li> <li><a href="/wiki/List_of_knot_theory_topics" title="List of knot theory topics">List of knot theory topics</a></li> <li><a href="/wiki/Molecular_knot" title="Molecular knot">Molecular knot</a></li> <li><a href="/wiki/Necktie#Knots" title="Necktie">Necktie § Knots</a></li> <li><a href="/wiki/Quantum_topology" title="Quantum topology">Quantum topology</a></li> <li><a href="/wiki/Ribbon_theory" class="mw-redirect" title="Ribbon theory">Ribbon theory</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=18" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAdams2004" class="citation cs2"><a href="/wiki/Colin_Adams_(mathematician)" title="Colin Adams (mathematician)">Adams, Colin</a> (2004), <i>The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots</i>, <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3678-1" title="Special:BookSources/978-0-8218-3678-1"><bdi>978-0-8218-3678-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Knot+Book%3A+An+Elementary+Introduction+to+the+Mathematical+Theory+of+Knots&rft.pub=American+Mathematical+Society&rft.date=2004&rft.isbn=978-0-8218-3678-1&rft.aulast=Adams&rft.aufirst=Colin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamsCrawfordDeMeoLandry2015" class="citation cs2">Adams, Colin; Crawford, Thomas; DeMeo, Benjamin; Landry, Michael; Lin, Alex Tong; Montee, MurphyKate; Park, Seojung; Venkatesh, Saraswathi; Yhee, Farrah (2015), "Knot projections with a single multi-crossing", <i>Journal of Knot Theory and Its Ramifications</i>, <b>24</b> (3): 1550011, 30, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1208.5742">1208.5742</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS021821651550011X">10.1142/S021821651550011X</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3342136">3342136</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119320887">119320887</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Knot+Theory+and+Its+Ramifications&rft.atitle=Knot+projections+with+a+single+multi-crossing&rft.volume=24&rft.issue=3&rft.pages=1550011%2C+30&rft.date=2015&rft_id=info%3Aarxiv%2F1208.5742&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3342136%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119320887%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS021821651550011X&rft.aulast=Adams&rft.aufirst=Colin&rft.au=Crawford%2C+Thomas&rft.au=DeMeo%2C+Benjamin&rft.au=Landry%2C+Michael&rft.au=Lin%2C+Alex+Tong&rft.au=Montee%2C+MurphyKate&rft.au=Park%2C+Seojung&rft.au=Venkatesh%2C+Saraswathi&rft.au=Yhee%2C+Farrah&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamsHildebrandWeeks1991" class="citation cs2">Adams, Colin; Hildebrand, Martin; <a href="/wiki/Jeffrey_Weeks_(mathematician)" title="Jeffrey Weeks (mathematician)">Weeks, Jeffrey</a> (1991), "Hyperbolic invariants of knots and links", <i>Transactions of the American Mathematical Society</i>, <b>326</b> (1): <span class="nowrap">1–</span>56, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1991-0994161-2">10.1090/s0002-9947-1991-0994161-2</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2001854">2001854</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Hyperbolic+invariants+of+knots+and+links&rft.volume=326&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E56&rft.date=1991&rft_id=info%3Adoi%2F10.1090%2Fs0002-9947-1991-0994161-2&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2001854%23id-name%3DJSTOR&rft.aulast=Adams&rft.aufirst=Colin&rft.au=Hildebrand%2C+Martin&rft.au=Weeks%2C+Jeffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAkbulutKing1981" class="citation cs2"><a href="/wiki/Selman_Akbulut" title="Selman Akbulut">Akbulut, Selman</a>; King, Henry C. (1981), "All knots are algebraic", <i><a href="/wiki/Comment._Math._Helv." class="mw-redirect" title="Comment. Math. Helv.">Comment. Math. Helv.</a></i>, <b>56</b> (3): <span class="nowrap">339–</span>351, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02566217">10.1007/BF02566217</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120218312">120218312</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comment.+Math.+Helv.&rft.atitle=All+knots+are+algebraic&rft.volume=56&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E339-%3C%2Fspan%3E351&rft.date=1981&rft_id=info%3Adoi%2F10.1007%2FBF02566217&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120218312%23id-name%3DS2CID&rft.aulast=Akbulut&rft.aufirst=Selman&rft.au=King%2C+Henry+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBar-Natan1995" class="citation cs2"><a href="/wiki/Dror_Bar-Natan" title="Dror Bar-Natan">Bar-Natan, Dror</a> (1995), "On the Vassiliev knot invariants", <i><a href="/wiki/Topology_(journal)" title="Topology (journal)">Topology</a></i>, <b>34</b> (2): <span class="nowrap">423–</span>472, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0040-9383%2895%2993237-2">10.1016/0040-9383(95)93237-2</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Topology&rft.atitle=On+the+Vassiliev+knot+invariants&rft.volume=34&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E423-%3C%2Fspan%3E472&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0040-9383%2895%2993237-2&rft.aulast=Bar-Natan&rft.aufirst=Dror&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurton2020" class="citation conference cs1">Burton, Benjamin A. (2020). <a rel="nofollow" class="external text" href="https://drops.dagstuhl.de/opus/volltexte/2020/12183/">"The Next 350 Million Knots"</a>. <i>36th International Symposium on Computational Geometry (SoCG 2020)</i>. Leibniz Int. Proc. Inform. Vol. 164. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. pp. 25:1–25:17. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4230%2FLIPIcs.SoCG.2020.25">10.4230/LIPIcs.SoCG.2020.25</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=The+Next+350+Million+Knots&rft.btitle=36th+International+Symposium+on+Computational+Geometry+%28SoCG+2020%29&rft.series=Leibniz+Int.+Proc.+Inform.&rft.pages=25%3A1-25%3A17&rft.pub=Schloss+Dagstuhl%E2%80%93Leibniz-Zentrum+f%C3%BCr+Informatik&rft.date=2020&rft_id=info%3Adoi%2F10.4230%2FLIPIcs.SoCG.2020.25&rft.aulast=Burton&rft.aufirst=Benjamin+A.&rft_id=https%3A%2F%2Fdrops.dagstuhl.de%2Fopus%2Fvolltexte%2F2020%2F12183%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCollins2006" class="citation cs2">Collins, Graham (April 2006), "Computing with Quantum Knots", <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>, <b>294</b> (4): <span class="nowrap">56–</span>63, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006SciAm.294d..56C">2006SciAm.294d..56C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0406-56">10.1038/scientificamerican0406-56</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16596880">16596880</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Computing+with+Quantum+Knots&rft.volume=294&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E56-%3C%2Fspan%3E63&rft.date=2006-04&rft_id=info%3Apmid%2F16596880&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0406-56&rft_id=info%3Abibcode%2F2006SciAm.294d..56C&rft.aulast=Collins&rft.aufirst=Graham&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDehn1914" class="citation cs2"><a href="/wiki/Max_Dehn" title="Max Dehn">Dehn, Max</a> (1914), "Die beiden Kleeblattschlingen", <i>Mathematische Annalen</i>, <b>75</b> (3): <span class="nowrap">402–</span>413, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01563732">10.1007/BF01563732</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120452571">120452571</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Die+beiden+Kleeblattschlingen&rft.volume=75&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E402-%3C%2Fspan%3E413&rft.date=1914&rft_id=info%3Adoi%2F10.1007%2FBF01563732&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120452571%23id-name%3DS2CID&rft.aulast=Dehn&rft.aufirst=Max&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConway1970" class="citation cs2"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John H.</a> (1970), "An enumeration of knots and links, and some of their algebraic properties", <i>Computational Problems in Abstract Algebra</i>, Pergamon, pp. <span class="nowrap">329–</span>358, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FB978-0-08-012975-4.50034-5">10.1016/B978-0-08-012975-4.50034-5</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-012975-4" title="Special:BookSources/978-0-08-012975-4"><bdi>978-0-08-012975-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+enumeration+of+knots+and+links%2C+and+some+of+their+algebraic+properties&rft.btitle=Computational+Problems+in+Abstract+Algebra&rft.pages=%3Cspan+class%3D%22nowrap%22%3E329-%3C%2Fspan%3E358&rft.pub=Pergamon&rft.date=1970&rft_id=info%3Adoi%2F10.1016%2FB978-0-08-012975-4.50034-5&rft.isbn=978-0-08-012975-4&rft.aulast=Conway&rft.aufirst=John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDollHoste1991" class="citation cs2">Doll, Helmut; Hoste, Jim (1991), "A tabulation of oriented links. With microfiche supplement", <i>Math. Comp.</i>, <b>57</b> (196): <span class="nowrap">747–</span>761, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991MaCom..57..747D">1991MaCom..57..747D</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1991-1094946-4">10.1090/S0025-5718-1991-1094946-4</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Comp.&rft.atitle=A+tabulation+of+oriented+links.+With+microfiche+supplement&rft.volume=57&rft.issue=196&rft.pages=%3Cspan+class%3D%22nowrap%22%3E747-%3C%2Fspan%3E761&rft.date=1991&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-1991-1094946-4&rft_id=info%3Abibcode%2F1991MaCom..57..747D&rft.aulast=Doll&rft.aufirst=Helmut&rft.au=Hoste%2C+Jim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlapan2000" class="citation cs2"><a href="/wiki/Erica_Flapan" title="Erica Flapan">Flapan, Erica</a> (2000), <a href="/wiki/When_Topology_Meets_Chemistry" title="When Topology Meets Chemistry"><i>When topology meets chemistry: A topological look at molecular chirality</i></a>, Outlook, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-66254-3" title="Special:BookSources/978-0-521-66254-3"><bdi>978-0-521-66254-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=When+topology+meets+chemistry%3A+A+topological+look+at+molecular+chirality&rft.series=Outlook&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-66254-3&rft.aulast=Flapan&rft.aufirst=Erica&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaefliger1962" class="citation cs2"><a href="/wiki/Andr%C3%A9_Haefliger" title="André Haefliger">Haefliger, André</a> (1962), "Knotted (4<i>k</i> − 1)-spheres in 6<i>k</i>-space", <i>Annals of Mathematics</i>, Second Series, <b>75</b> (3): <span class="nowrap">452–</span>466, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970208">10.2307/1970208</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970208">1970208</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Knotted+%284k+%26minus%3B+1%29-spheres+in+6k-space&rft.volume=75&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E452-%3C%2Fspan%3E466&rft.date=1962&rft_id=info%3Adoi%2F10.2307%2F1970208&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970208%23id-name%3DJSTOR&rft.aulast=Haefliger&rft.aufirst=Andr%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaken1962" class="citation cs2"><a href="/wiki/Wolfgang_Haken" title="Wolfgang Haken">Haken, Wolfgang</a> (1962), "Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I", <i><a href="/wiki/Mathematische_Zeitschrift" title="Mathematische Zeitschrift">Mathematische Zeitschrift</a></i>, <b>80</b>: <span class="nowrap">89–</span>120, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01162369">10.1007/BF01162369</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0160196">0160196</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=%C3%9Cber+das+Hom%C3%B6omorphieproblem+der+3-Mannigfaltigkeiten.+I&rft.volume=80&rft.pages=%3Cspan+class%3D%22nowrap%22%3E89-%3C%2Fspan%3E120&rft.date=1962&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0160196%23id-name%3DMR&rft.issn=0025-5874&rft_id=info%3Adoi%2F10.1007%2FBF01162369&rft.aulast=Haken&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHass1998" class="citation cs2"><a href="/wiki/Joel_Hass" title="Joel Hass">Hass, Joel</a> (1998), "Algorithms for recognizing knots and 3-manifolds", <i><a href="/wiki/Chaos,_Solitons_and_Fractals" class="mw-redirect" title="Chaos, Solitons and Fractals">Chaos, Solitons and Fractals</a></i>, <b>9</b> (<span class="nowrap">4–</span>5): <span class="nowrap">569–</span>581, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9712269">math/9712269</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998CSF.....9..569H">1998CSF.....9..569H</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0960-0779%2897%2900109-4">10.1016/S0960-0779(97)00109-4</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7381505">7381505</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chaos%2C+Solitons+and+Fractals&rft.atitle=Algorithms+for+recognizing+knots+and+3-manifolds&rft.volume=9&rft.issue=%3Cspan+class%3D%22nowrap%22%3E4%E2%80%93%3C%2Fspan%3E5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E569-%3C%2Fspan%3E581&rft.date=1998&rft_id=info%3Aarxiv%2Fmath%2F9712269&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7381505%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2FS0960-0779%2897%2900109-4&rft_id=info%3Abibcode%2F1998CSF.....9..569H&rft.aulast=Hass&rft.aufirst=Joel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHosteThistlethwaiteWeeks1998" class="citation cs2">Hoste, Jim; <a href="/wiki/Morwen_Thistlethwaite" title="Morwen Thistlethwaite">Thistlethwaite, Morwen</a>; Weeks, Jeffrey (1998), "The First 1,701,935 Knots", <i><a href="/wiki/Math._Intelligencer" class="mw-redirect" title="Math. Intelligencer">Math. Intelligencer</a></i>, <b>20</b> (4): <span class="nowrap">33–</span>48, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03025227">10.1007/BF03025227</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18027155">18027155</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Intelligencer&rft.atitle=The+First+1%2C701%2C935+Knots&rft.volume=20&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E33-%3C%2Fspan%3E48&rft.date=1998&rft_id=info%3Adoi%2F10.1007%2FBF03025227&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18027155%23id-name%3DS2CID&rft.aulast=Hoste&rft.aufirst=Jim&rft.au=Thistlethwaite%2C+Morwen&rft.au=Weeks%2C+Jeffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoste2005" class="citation book cs1">Hoste, Jim (2005). "The Enumeration and Classification of Knots and Links". <i>Handbook of Knot Theory</i>. pp. <span class="nowrap">209–</span>232. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FB978-044451452-3%2F50006-X">10.1016/B978-044451452-3/50006-X</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-51452-3" title="Special:BookSources/978-0-444-51452-3"><bdi>978-0-444-51452-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Enumeration+and+Classification+of+Knots+and+Links&rft.btitle=Handbook+of+Knot+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E209-%3C%2Fspan%3E232&rft.date=2005&rft_id=info%3Adoi%2F10.1016%2FB978-044451452-3%2F50006-X&rft.isbn=978-0-444-51452-3&rft.aulast=Hoste&rft.aufirst=Jim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevine1965" class="citation cs2"><a href="/wiki/Jerome_Levine" title="Jerome Levine">Levine, Jerome</a> (1965), "A classification of differentiable knots", <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, Second Series, <b>1982</b> (1): <span class="nowrap">15–</span>50, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970561">10.2307/1970561</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970561">1970561</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=A+classification+of+differentiable+knots&rft.volume=1982&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E15-%3C%2Fspan%3E50&rft.date=1965&rft_id=info%3Adoi%2F10.2307%2F1970561&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970561%23id-name%3DJSTOR&rft.aulast=Levine&rft.aufirst=Jerome&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKontsevich1993" class="citation book cs1">Kontsevich, M. (1993). "Vassiliev's knot invariants". <i>I. M. Gelfand Seminar</i>. ADVSOV. Vol. 16. pp. <span class="nowrap">137–</span>150. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fadvsov%2F016.2%2F04">10.1090/advsov/016.2/04</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4117-4" title="Special:BookSources/978-0-8218-4117-4"><bdi>978-0-8218-4117-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Vassiliev%27s+knot+invariants&rft.btitle=I.+M.+Gelfand+Seminar&rft.series=ADVSOV&rft.pages=%3Cspan+class%3D%22nowrap%22%3E137-%3C%2Fspan%3E150&rft.date=1993&rft_id=info%3Adoi%2F10.1090%2Fadvsov%2F016.2%2F04&rft.isbn=978-0-8218-4117-4&rft.aulast=Kontsevich&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLickorish1997" class="citation cs2"><a href="/wiki/W._B._R._Lickorish" title="W. B. R. Lickorish">Lickorish, W. B. Raymond</a> (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PhHhw_kRvewC"><i>An Introduction to Knot Theory</i></a>, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0691-0">10.1007/978-1-4612-0691-0</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98254-0" title="Special:BookSources/978-0-387-98254-0"><bdi>978-0-387-98254-0</bdi></a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122824389">122824389</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Knot+Theory&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1997&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122824389%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-0691-0&rft.isbn=978-0-387-98254-0&rft.aulast=Lickorish&rft.aufirst=W.+B.+Raymond&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPhHhw_kRvewC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerko1974" class="citation cs2">Perko, Kenneth (1974), "On the classification of knots", <i><a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a></i>, <b>45</b> (2): <span class="nowrap">262–</span>6, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2040074">10.2307/2040074</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2040074">2040074</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=On+the+classification+of+knots&rft.volume=45&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E262-%3C%2Fspan%3E6&rft.date=1974&rft_id=info%3Adoi%2F10.2307%2F2040074&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2040074%23id-name%3DJSTOR&rft.aulast=Perko&rft.aufirst=Kenneth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRolfsen1976" class="citation cs2">Rolfsen, Dale (1976), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=naYJBAAAQBAJ"><i>Knots and Links</i></a>, Mathematics Lecture Series, vol. 7, <a href="/wiki/Berkeley,_California" title="Berkeley, California">Berkeley, California</a>: Publish or Perish, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-914098-16-4" title="Special:BookSources/978-0-914098-16-4"><bdi>978-0-914098-16-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0515288">0515288</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots+and+Links&rft.place=Berkeley%2C+California&rft.series=Mathematics+Lecture+Series&rft.pub=Publish+or+Perish&rft.date=1976&rft.isbn=978-0-914098-16-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0515288%23id-name%3DMR&rft.aulast=Rolfsen&rft.aufirst=Dale&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnaYJBAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchubert1949" class="citation book cs1">Schubert, Horst (1949). <i>Die eindeutige Zerlegbarkeit eines Knotens in Primknoten</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-45813-2">10.1007/978-3-642-45813-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-01419-5" title="Special:BookSources/978-3-540-01419-5"><bdi>978-3-540-01419-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+eindeutige+Zerlegbarkeit+eines+Knotens+in+Primknoten&rft.date=1949&rft_id=info%3Adoi%2F10.1007%2F978-3-642-45813-2&rft.isbn=978-3-540-01419-5&rft.aulast=Schubert&rft.aufirst=Horst&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilver2006" class="citation journal cs1">Silver, Daniel (2006). "Knot Theory's Odd Origins". <i>American Scientist</i>. <b>94</b> (2): 158. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1511%2F2006.2.158">10.1511/2006.2.158</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Scientist&rft.atitle=Knot+Theory%27s+Odd+Origins&rft.volume=94&rft.issue=2&rft.pages=158&rft.date=2006&rft_id=info%3Adoi%2F10.1511%2F2006.2.158&rft.aulast=Silver&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimon1986" class="citation cs2">Simon, Jonathan (1986), "Topological chirality of certain molecules", <i>Topology</i>, <b>25</b> (2): <span class="nowrap">229–</span>235, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0040-9383%2886%2990041-8">10.1016/0040-9383(86)90041-8</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Topology&rft.atitle=Topological+chirality+of+certain+molecules&rft.volume=25&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E229-%3C%2Fspan%3E235&rft.date=1986&rft_id=info%3Adoi%2F10.1016%2F0040-9383%2886%2990041-8&rft.aulast=Simon&rft.aufirst=Jonathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSossinsky2002" class="citation cs2">Sossinsky, Alexei (2002), <i>Knots, mathematics with a twist</i>, Harvard University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-00944-8" title="Special:BookSources/978-0-674-00944-8"><bdi>978-0-674-00944-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots%2C+mathematics+with+a+twist&rft.pub=Harvard+University+Press&rft.date=2002&rft.isbn=978-0-674-00944-8&rft.aulast=Sossinsky&rft.aufirst=Alexei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuraev2016" class="citation book cs1">Turaev, Vladimir G. (2016). <i>Quantum Invariants of Knots and 3-Manifolds</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783110435221">10.1515/9783110435221</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-043522-1" title="Special:BookSources/978-3-11-043522-1"><bdi>978-3-11-043522-1</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118682559">118682559</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Invariants+of+Knots+and+3-Manifolds&rft.date=2016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118682559%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1515%2F9783110435221&rft.isbn=978-3-11-043522-1&rft.aulast=Turaev&rft.aufirst=Vladimir+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein2013" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> (2013). <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ReducedKnotDiagram.html">"Reduced Knot Diagram"</a>. <i>MathWorld</i>. Wolfram<span class="reference-accessdate">. Retrieved <span class="nowrap">8 May</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Reduced+Knot+Diagram&rft.date=2013&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FReducedKnotDiagram.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein2013a" class="citation web cs1">Weisstein, Eric W. (2013a). <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ReducibleCrossing.html">"Reducible Crossing"</a>. <i>MathWorld</i>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram</a><span class="reference-accessdate">. Retrieved <span class="nowrap">8 May</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Reducible+Crossing&rft.date=2013&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FReducibleCrossing.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWitten1989" class="citation cs2"><a href="/wiki/Edward_Witten" title="Edward Witten">Witten, Edward</a> (1989), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.cmp/1104178138">"Quantum field theory and the Jones polynomial"</a>, <i>Comm. Math. Phys.</i>, <b>121</b> (3): <span class="nowrap">351–</span>399, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989CMaPh.121..351W">1989CMaPh.121..351W</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01217730">10.1007/BF01217730</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14951363">14951363</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comm.+Math.+Phys.&rft.atitle=Quantum+field+theory+and+the+Jones+polynomial&rft.volume=121&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E351-%3C%2Fspan%3E399&rft.date=1989&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14951363%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01217730&rft_id=info%3Abibcode%2F1989CMaPh.121..351W&rft.aulast=Witten&rft.aufirst=Edward&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.cmp%2F1104178138&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZeeman1963" class="citation cs2"><a href="/wiki/E._C._Zeeman" class="mw-redirect" title="E. C. Zeeman">Zeeman, Erik C.</a> (1963), "Unknotting combinatorial balls", <i>Annals of Mathematics</i>, Second Series, <b>78</b> (3): <span class="nowrap">501–</span>526, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970538">10.2307/1970538</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970538">1970538</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Unknotting+combinatorial+balls&rft.volume=78&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E501-%3C%2Fspan%3E526&rft.date=1963&rft_id=info%3Adoi%2F10.2307%2F1970538&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970538%23id-name%3DJSTOR&rft.aulast=Zeeman&rft.aufirst=Erik+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=19" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">As first sketched using the theory of <a href="/wiki/Haken_manifold" title="Haken manifold">Haken manifolds</a> by <a href="#CITEREFHaken1962">Haken (1962)</a>. For a more recent survey, see <a href="#CITEREFHass1998">Hass (1998)</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.maths.ox.ac.uk/node/38304"><i>Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time</i></a>, Mathematical Institute, <a href="/wiki/University_of_Oxford" title="University of Oxford">University of Oxford</a>, 2021-02-03<span class="reference-accessdate">, retrieved <span class="nowrap">2021-02-03</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Marc+Lackenby+announces+a+new+unknot+recognition+algorithm+that+runs+in+quasi-polynomial+time&rft.pub=Mathematical+Institute%2C+University+of+Oxford&rft.date=2021-02-03&rft_id=https%3A%2F%2Fwww.maths.ox.ac.uk%2Fnode%2F38304&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEWeisstein2013-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeisstein2013_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeisstein2013">Weisstein 2013</a>.</span> </li> <li id="cite_note-FOOTNOTEWeisstein2013a-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeisstein2013a_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeisstein2013a">Weisstein 2013a</a>.</span> </li> <li id="cite_note-FOOTNOTEAdamsCrawfordDeMeoLandry2015-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAdamsCrawfordDeMeoLandry2015_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAdamsCrawfordDeMeoLandry2015">Adams et al. 2015</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevineOrr2000" class="citation cs2">Levine, J.; Orr, K (2000), "A survey of applications of surgery to knot and link theory", <i>Surveys on Surgery Theory: Papers Dedicated to C.T.C. Wall</i>, Annals of mathematics studies, vol. 1, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.4359">10.1.1.64.4359</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0691049380" title="Special:BookSources/978-0691049380"><bdi>978-0691049380</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+survey+of+applications+of+surgery+to+knot+and+link+theory&rft.btitle=Surveys+on+Surgery+Theory%3A+Papers+Dedicated+to+C.T.C.+Wall&rft.series=Annals+of+mathematics+studies&rft.pub=Princeton+University+Press&rft.date=2000&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.64.4359%23id-name%3DCiteSeerX&rft.isbn=978-0691049380&rft.aulast=Levine&rft.aufirst=J.&rft.au=Orr%2C+K&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span> — An introductory article to high dimensional knots and links for the advanced readers</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOgasa2013" class="citation cs2">Ogasa, Eiji (2013), <i>Introduction to high dimensional knots</i>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1304.6053">1304.6053</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013arXiv1304.6053O">2013arXiv1304.6053O</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+high+dimensional+knots&rft.date=2013&rft_id=info%3Aarxiv%2F1304.6053&rft_id=info%3Abibcode%2F2013arXiv1304.6053O&rft.aulast=Ogasa&rft.aufirst=Eiji&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span> — An introductory article to high dimensional knots and links for beginners</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolovnevMashaghi2021" class="citation journal cs1">Golovnev, Anatoly; Mashaghi, Alireza (7 December 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsym13122353">"Circuit Topology for Bottom-Up Engineering of Molecular Knots"</a>. <i>Symmetry</i>. <b>13</b> (12): 2353. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2106.03925">2106.03925</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021Symm...13.2353G">2021Symm...13.2353G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsym13122353">10.3390/sym13122353</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Symmetry&rft.atitle=Circuit+Topology+for+Bottom-Up+Engineering+of+Molecular+Knots&rft.volume=13&rft.issue=12&rft.pages=2353&rft.date=2021-12-07&rft_id=info%3Aarxiv%2F2106.03925&rft_id=info%3Adoi%2F10.3390%2Fsym13122353&rft_id=info%3Abibcode%2F2021Symm...13.2353G&rft.aulast=Golovnev&rft.aufirst=Anatoly&rft.au=Mashaghi%2C+Alireza&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fsym13122353&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlapanMashaghiWong2023" class="citation journal cs1">Flapan, Erica; Mashaghi, Alireza; Wong, Helen (1 June 2023). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10235088">"A tile model of circuit topology for self-entangled biopolymers"</a>. <i>Scientific Reports</i>. <b>13</b> (1): 8889. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023NatSR..13.8889F">2023NatSR..13.8889F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41598-023-35771-8">10.1038/s41598-023-35771-8</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10235088">10235088</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/37264056">37264056</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=A+tile+model+of+circuit+topology+for+self-entangled+biopolymers&rft.volume=13&rft.issue=1&rft.pages=8889&rft.date=2023-06-01&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10235088%23id-name%3DPMC&rft_id=info%3Apmid%2F37264056&rft_id=info%3Adoi%2F10.1038%2Fs41598-023-35771-8&rft_id=info%3Abibcode%2F2023NatSR..13.8889F&rft.aulast=Flapan&rft.aufirst=Erica&rft.au=Mashaghi%2C+Alireza&rft.au=Wong%2C+Helen&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10235088&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://richardelwes.co.uk/2013/08/14/the-revenge-of-the-perko-pair/">The Revenge of the Perko Pair</a>", <i>RichardElwes.co.uk</i>. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=20" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Introductory_textbooks">Introductory textbooks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=21" title="Edit section: Introductory textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is (<a href="#CITEREFRolfsen1976">Rolfsen 1976</a>). Other good texts from the references are (<a href="#CITEREFAdams2004">Adams 2004</a>) and (<a href="#CITEREFLickorish1997">Lickorish 1997</a>). Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics. (<a href="#CITEREFCromwell2004">Cromwell 2004</a>) is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required. </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurdeZieschang1985" class="citation cs2"><a href="/w/index.php?title=Gerhard_Burde&action=edit&redlink=1" class="new" title="Gerhard Burde (page does not exist)">Burde, Gerhard</a>; <a href="/wiki/Heiner_Zieschang" title="Heiner Zieschang">Zieschang, Heiner</a> (1985), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DJHI7DpgIbIC"><i>Knots</i></a>, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-008675-1" title="Special:BookSources/978-3-11-008675-1"><bdi>978-3-11-008675-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots&rft.series=De+Gruyter+Studies+in+Mathematics&rft.pub=Walter+de+Gruyter&rft.date=1985&rft.isbn=978-3-11-008675-1&rft.aulast=Burde&rft.aufirst=Gerhard&rft.au=Zieschang%2C+Heiner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDJHI7DpgIbIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrowellFox1977" class="citation book cs1"><a href="/w/index.php?title=Richard_H._Crowell&action=edit&redlink=1" class="new" title="Richard H. Crowell (page does not exist)">Crowell, Richard H.</a>; <a href="/wiki/Ralph_Fox" title="Ralph Fox">Fox, Ralph</a> (1977). <i>Introduction to Knot Theory</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90272-2" title="Special:BookSources/978-0-387-90272-2"><bdi>978-0-387-90272-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Knot+Theory&rft.pub=Springer&rft.date=1977&rft.isbn=978-0-387-90272-2&rft.aulast=Crowell&rft.aufirst=Richard+H.&rft.au=Fox%2C+Ralph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKauffman1987" class="citation cs2"><a href="/wiki/Louis_H._Kauffman" class="mw-redirect" title="Louis H. Kauffman">Kauffman, Louis H.</a> (1987), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BLvGkIY8YzwC"><i>On Knots</i></a>, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08435-0" title="Special:BookSources/978-0-691-08435-0"><bdi>978-0-691-08435-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+Knots&rft.pub=Princeton+University+Press&rft.date=1987&rft.isbn=978-0-691-08435-0&rft.aulast=Kauffman&rft.aufirst=Louis+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBLvGkIY8YzwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKauffman2013" class="citation cs2"><a href="/wiki/Louis_H._Kauffman" class="mw-redirect" title="Louis H. Kauffman">Kauffman, Louis H.</a> (2013), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fSKrRQ77FMkC"><i>Knots and Physics</i></a> (4th ed.), World Scientific, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4383-00-4" title="Special:BookSources/978-981-4383-00-4"><bdi>978-981-4383-00-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots+and+Physics&rft.edition=4th&rft.pub=World+Scientific&rft.date=2013&rft.isbn=978-981-4383-00-4&rft.aulast=Kauffman&rft.aufirst=Louis+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfSKrRQ77FMkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCromwell2004" class="citation cs2">Cromwell, Peter R. (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=djvbTNR2dCwC"><i>Knots and Links</i></a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-54831-1" title="Special:BookSources/978-0-521-54831-1"><bdi>978-0-521-54831-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots+and+Links&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-0-521-54831-1&rft.aulast=Cromwell&rft.aufirst=Peter+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdjvbTNR2dCwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Surveys">Surveys</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=22" title="Edit section: Surveys"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMenascoThistlethwaite2005" class="citation cs2">Menasco, William W.; <a href="/wiki/Morwen_Thistlethwaite" title="Morwen Thistlethwaite">Thistlethwaite, Morwen</a>, eds. (2005), <i>Handbook of Knot Theory</i>, Elsevier, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-51452-3" title="Special:BookSources/978-0-444-51452-3"><bdi>978-0-444-51452-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Knot+Theory&rft.pub=Elsevier&rft.date=2005&rft.isbn=978-0-444-51452-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span> <ul><li>Menasco and Thistlethwaite's handbook surveys a mix of topics relevant to current research trends in a manner accessible to advanced undergraduates but of interest to professional researchers.</li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLivio2009" class="citation cs2">Livio, Mario (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ebd6QofqY6QC&pg=PA203">"Ch. 8: Unreasonable Effectiveness?"</a>, <i>Is God a Mathematician?</i>, Simon & Schuster, pp. <span class="nowrap">203–</span>218, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7432-9405-8" title="Special:BookSources/978-0-7432-9405-8"><bdi>978-0-7432-9405-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ch.+8%3A+Unreasonable+Effectiveness%3F&rft.btitle=Is+God+a+Mathematician%3F&rft.pages=%3Cspan+class%3D%22nowrap%22%3E203-%3C%2Fspan%3E218&rft.pub=Simon+%26+Schuster&rft.date=2009&rft.isbn=978-0-7432-9405-8&rft.aulast=Livio&rft.aufirst=Mario&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Debd6QofqY6QC%26pg%3DPA203&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Knot_theory" class="extiw" title="commons:Category:Knot theory">Knot theory</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/knot_theory" class="extiw" title="wiktionary:Special:Search/knot theory">knot theory</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.groupoids.org.uk/popmath/cpm/exhib/knotexhib.html">"Mathematics and Knots"</a> This is an online version of an exhibition developed for the 1989 Royal Society "PopMath RoadShow". Its aim was to use knots to present methods of mathematics to the general public.</li></ul> <div class="mw-heading mw-heading3"><h3 id="History_2">History</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=24" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomson1867" class="citation cs2"><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Thomson, Sir William</a> (1867), <a rel="nofollow" class="external text" href="http://zapatopi.net/kelvin/papers/on_vortex_atoms.html">"On Vortex Atoms"</a>, <i>Proceedings of the Royal Society of Edinburgh</i>, <b>VI</b>: <span class="nowrap">94–</span>105</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+Edinburgh&rft.atitle=On+Vortex+Atoms&rft.volume=VI&rft.pages=%3Cspan+class%3D%22nowrap%22%3E94-%3C%2Fspan%3E105&rft.date=1867&rft.aulast=Thomson&rft.aufirst=Sir+William&rft_id=http%3A%2F%2Fzapatopi.net%2Fkelvin%2Fpapers%2Fon_vortex_atoms.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilliman1963" class="citation cs2">Silliman, Robert H. (December 1963), "William Thomson: Smoke Rings and Nineteenth-Century Atomism", <i>Isis</i>, <b>54</b> (4): <span class="nowrap">461–</span>474, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F349764">10.1086/349764</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/228151">228151</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:144988108">144988108</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Isis&rft.atitle=William+Thomson%3A+Smoke+Rings+and+Nineteenth-Century+Atomism&rft.volume=54&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E461-%3C%2Fspan%3E474&rft.date=1963-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A144988108%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F228151%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1086%2F349764&rft.aulast=Silliman&rft.aufirst=Robert+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKnot+theory" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://jagworks.southalabama.edu/knot-theory-hist/4/">Movie</a> of a modern recreation of Tait's smoke ring experiment</li> <li><a rel="nofollow" class="external text" href="http://www.maths.ed.ac.uk/~aar/knots">History of knot theory</a> (on the home page of <a href="/wiki/Andrew_Ranicki" title="Andrew Ranicki">Andrew Ranicki</a>)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Knot_tables_and_software">Knot tables and software</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Knot_theory&action=edit&section=25" title="Edit section: Knot tables and software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://knotinfo.math.indiana.edu/"><b>KnotInfo</b>: <i>Table of Knot Invariants and Knot Theory Resources</i></a></li> <li><a rel="nofollow" class="external text" href="http://katlas.math.toronto.edu/wiki/Main_Page">The Knot Atlas</a> — detailed info on individual knots in knot tables</li> <li><a rel="nofollow" class="external text" href="http://knotplot.com/">KnotPlot</a> — software to investigate geometric properties of knots</li> <li><a rel="nofollow" class="external text" href="http://www.math.utk.edu/~morwen/knotscape.html">Knotscape</a> — software to create images of knots</li> <li><a rel="nofollow" class="external text" href="http://knotilus.math.uwo.ca/">Knoutilus</a> — online database and image generator of knots</li> <li><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/KnotData.html">KnotData.html</a> — <a href="/wiki/Wolfram_Mathematica" title="Wolfram Mathematica">Wolfram Mathematica</a> function for investigating knots</li> <li><a rel="nofollow" class="external text" href="https://regina-normal.github.io/index.html/">Regina</a> — software for low-dimensional topology with native support for knots and links. <a rel="nofollow" class="external text" href="https://regina-normal.github.io/data.html#knots">Tables</a> of prime knots with up to 19 crossings</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist 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href="/wiki/Template:Knot_theory" title="Template:Knot theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Knot_theory" title="Template talk:Knot theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Knot_theory" title="Special:EditPage/Template:Knot theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Knot_theory_(knots_and_links)479" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Knot theory</a> (<a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knots</a> and <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">links</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hyperbolic_link" title="Hyperbolic link">Hyperbolic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Figure-eight_knot_(mathematics)" title="Figure-eight knot (mathematics)">Figure-eight</a> (4<sub>1</sub>)</li> <li><a href="/wiki/Three-twist_knot" title="Three-twist knot">Three-twist</a> (5<sub>2</sub>)</li> <li><a href="/wiki/Stevedore_knot_(mathematics)" title="Stevedore knot (mathematics)">Stevedore</a> (6<sub>1</sub>)</li> <li><a href="/wiki/6%E2%82%82_knot" class="mw-redirect" title="6₂ knot">6<sub>2</sub></a></li> <li><a href="/wiki/6%E2%82%83_knot" class="mw-redirect" title="6₃ knot">6<sub>3</sub></a></li> <li><a href="/wiki/7%E2%82%84_knot" class="mw-redirect" title="7₄ knot">Endless</a> (7<sub>4</sub>)</li> <li><a href="/wiki/Carrick_mat" title="Carrick mat">Carrick mat</a> (8<sub>18</sub>)</li> <li><a href="/wiki/Perko_pair" title="Perko pair">Perko pair</a> (10<sub>161</sub>)</li> <li><a href="/wiki/Conway_knot" title="Conway knot">Conway knot</a> (11n34)</li> <li><a href="/wiki/Kinoshita%E2%80%93Terasaka_knot" title="Kinoshita–Terasaka knot">Kinoshita–Terasaka knot</a> (11n42)</li> <li><a href="/wiki/(%E2%88%922,3,7)_pretzel_knot" title="(−2,3,7) pretzel knot">(−2,3,7) pretzel</a> (12n242)</li> <li><a href="/wiki/Whitehead_link" title="Whitehead link">Whitehead</a> (5<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a> (6<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">3</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span>)</li> <li><a href="/wiki/L10a140_link" title="L10a140 link">L10a140</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Satellite_knot" title="Satellite knot">Satellite</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composite_knot" class="mw-redirect" title="Composite knot">Composite knots</a> <ul><li><a href="/wiki/Granny_knot_(mathematics)" title="Granny knot (mathematics)">Granny</a></li> <li><a href="/wiki/Square_knot_(mathematics)" title="Square knot (mathematics)">Square</a></li></ul></li> <li><a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Knot sum</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Torus_knot" title="Torus knot">Torus</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Unknot" title="Unknot">Unknot</a> (0<sub>1</sub>)</li> <li><a href="/wiki/Trefoil_knot" title="Trefoil knot">Trefoil</a> (3<sub>1</sub>)</li> <li><a href="/wiki/Cinquefoil_knot" title="Cinquefoil knot">Cinquefoil</a> (5<sub>1</sub>)</li> <li><a href="/wiki/7%E2%82%81_knot" class="mw-redirect" title="7₁ knot">Septafoil</a> (7<sub>1</sub>)</li> <li><a href="/wiki/Unlink" title="Unlink">Unlink</a> (0<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Hopf_link" title="Hopf link">Hopf</a> (2<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Solomon%27s_knot" title="Solomon's knot">Solomon's</a> (4<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Knot_invariant" title="Knot invariant">Invariants</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternating_knot" title="Alternating knot">Alternating</a></li> <li><a href="/wiki/Arf_invariant_of_a_knot" title="Arf invariant of a knot">Arf invariant</a></li> <li><a href="/wiki/Bridge_number" title="Bridge number">Bridge no.</a> <ul><li><a href="/wiki/2-bridge_knot" title="2-bridge knot">2-bridge</a></li></ul></li> <li><a href="/wiki/Brunnian_link" title="Brunnian link">Brunnian</a></li> <li><a href="/wiki/Chiral_knot" title="Chiral knot">Chirality</a> <ul><li><a href="/wiki/Invertible_knot" title="Invertible knot">Invertible</a></li></ul></li> <li><a href="/wiki/Crosscap_number" title="Crosscap number">Crosscap no.</a></li> <li><a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">Crossing no.</a></li> <li><a href="/wiki/Finite_type_invariant" title="Finite type invariant">Finite type invariant</a></li> <li><a href="/wiki/Hyperbolic_volume" title="Hyperbolic volume">Hyperbolic volume</a></li> <li><a href="/wiki/Khovanov_homology" title="Khovanov homology">Khovanov homology</a></li> <li><a href="/wiki/Knot_genus" class="mw-redirect" title="Knot genus">Genus</a></li> <li><a href="/wiki/Knot_group" title="Knot group">Knot group</a></li> <li><a href="/wiki/Link_group" title="Link group">Link group</a></li> <li><a href="/wiki/Linking_number" title="Linking number">Linking no.</a></li> <li><a href="/wiki/Knot_polynomial" title="Knot polynomial">Polynomial</a> <ul><li><a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander</a></li> <li><a href="/wiki/Bracket_polynomial" title="Bracket polynomial">Bracket</a></li> <li><a href="/wiki/HOMFLY_polynomial" title="HOMFLY polynomial">HOMFLY</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones</a></li> <li><a href="/wiki/Kauffman_polynomial" title="Kauffman polynomial">Kauffman</a></li></ul></li> <li><a href="/wiki/Pretzel_link" title="Pretzel link">Pretzel</a></li> <li><a href="/wiki/Prime_knot" title="Prime knot">Prime</a> <ul><li><a href="/wiki/List_of_prime_knots" title="List of prime knots">list</a></li></ul></li> <li><a href="/wiki/Stick_number" title="Stick number">Stick no.</a></li> <li><a href="/wiki/Tricolorability" title="Tricolorability">Tricolorability</a></li> <li><a href="/wiki/Unknotting_number" title="Unknotting number">Unknotting no.</a> and <a href="/wiki/Unknotting_problem" title="Unknotting problem">problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation<br />and <a href="/wiki/Knot_operation" title="Knot operation">operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander%E2%80%93Briggs_notation" class="mw-redirect" title="Alexander–Briggs notation">Alexander–Briggs notation</a></li> <li><a href="/wiki/Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation</a></li> <li><a href="/wiki/Dowker%E2%80%93Thistlethwaite_notation" title="Dowker–Thistlethwaite notation">Dowker–Thistlethwaite notation</a></li> <li><a href="/wiki/Flype" title="Flype">Flype</a></li> <li><a href="/wiki/Mutation_(knot_theory)" title="Mutation (knot theory)">Mutation</a></li> <li><a href="/wiki/Reidemeister_move" title="Reidemeister move">Reidemeister move</a></li> <li><a href="/wiki/Skein_relation" title="Skein relation">Skein relation</a></li> <li><a href="/wiki/Knot_tabulation" title="Knot tabulation">Tabulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander%27s_theorem" title="Alexander's theorem">Alexander's theorem</a></li> <li><a href="/wiki/Berge_knot" title="Berge knot">Berge</a></li> <li><a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">Braid theory</a></li> <li><a href="/wiki/Conway_sphere" title="Conway sphere">Conway sphere</a></li> <li><a href="/wiki/Knot_complement" title="Knot complement">Complement</a></li> <li><a href="/wiki/Double_torus_knot" class="mw-redirect" title="Double torus knot">Double torus</a></li> <li><a href="/wiki/Fibered_knot" title="Fibered knot">Fibered</a></li> <li><a href="/wiki/Knot" title="Knot">Knot</a></li> <li><a href="/wiki/List_of_mathematical_knots_and_links" title="List of mathematical knots and links">List of knots and links</a></li> <li><a href="/wiki/Ribbon_knot" title="Ribbon knot">Ribbon</a></li> <li><a href="/wiki/Slice_knot" title="Slice knot">Slice</a></li> <li><a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Sum</a></li> <li><a href="/wiki/Tait_conjectures" title="Tait conjectures">Tait conjectures</a></li> <li><a href="/wiki/Twist_knot" title="Twist knot">Twist</a></li> <li><a href="/wiki/Wild_knot" title="Wild knot">Wild</a></li> <li><a href="/wiki/Writhe" title="Writhe">Writhe</a></li> <li><a href="/wiki/Surgery_theory" title="Surgery theory">Surgery theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Knot_theory" title="Category:Knot theory">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" 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mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-b766959bd-89cpd","wgBackendResponseTime":178,"wgPageParseReport":{"limitreport":{"cputime":"1.350","walltime":"1.705","ppvisitednodes":{"value":3987,"limit":1000000},"postexpandincludesize":{"value":129063,"limit":2097152},"templateargumentsize":{"value":1073,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":11,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":130105,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1288.525 1 -total"," 35.77% 460.947 33 Template:Citation"," 11.00% 141.717 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