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class="vector-toc-link" href="#g-torus_knot"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span><i>g</i>-torus knot</span> </div> </a> <ul id="toc-g-torus_knot-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </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">7</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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </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">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </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" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Torus_link&redirect=no" class="mw-redirect" title="Torus link">Torus link</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Knot which lies on the surface of a torus in 3-dimensional space</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">Not to be confused with <a href="/wiki/Torus" title="Torus">Torus</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TorusKnot3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/TorusKnot3D.png/220px-TorusKnot3D.png" decoding="async" width="220" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/TorusKnot3D.png/330px-TorusKnot3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/07/TorusKnot3D.png 2x" data-file-width="375" data-file-height="355" /></a><figcaption>A (3,−7)-<a href="/wiki/3D_computer_graphics" title="3D computer graphics">3D</a> torus knot.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Eurelea.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/Eurelea.png" decoding="async" width="220" height="244" class="mw-file-element" data-file-width="220" data-file-height="244" /></a><figcaption><a href="/w/index.php?title=EureleA&action=edit&redlink=1" class="new" title="EureleA (page does not exist)">EureleA</a> Award showing a (2,3)-torus knot.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:(2,8)-Torus_Link.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/%282%2C8%29-Torus_Link.svg/220px-%282%2C8%29-Torus_Link.svg.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/%282%2C8%29-Torus_Link.svg/330px-%282%2C8%29-Torus_Link.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/%282%2C8%29-Torus_Link.svg/440px-%282%2C8%29-Torus_Link.svg.png 2x" data-file-width="232" data-file-height="233" /></a><figcaption>(2,8) torus link</figcaption></figure> <p>In <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>, a <b>torus knot</b> is a special kind of <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot</a> that lies on the surface of an unknotted <a href="/wiki/Torus" title="Torus">torus</a> in <b>R</b><sup>3</sup>. Similarly, a <b>torus link</b> is a <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">link</a> which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> <a href="/wiki/Integer" title="Integer">integers</a> <i>p</i> and <i>q</i>. A torus link arises if <i>p</i> and <i>q</i> are not coprime (in which case the number of components is <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">gcd</a>(<i>p, q</i>)). A torus knot is <a href="/wiki/Unknot" title="Unknot">trivial</a> (equivalent to the unknot) <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> either <i>p</i> or <i>q</i> is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knot</a>. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Trefoil_knot_left.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Trefoil_knot_left.svg/220px-Trefoil_knot_left.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Trefoil_knot_left.svg/330px-Trefoil_knot_left.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Trefoil_knot_left.svg/440px-Trefoil_knot_left.svg.png 2x" data-file-width="250" data-file-height="250" /></a><figcaption>the (2,−3)-torus knot, also known as the left-handed <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knot</a></figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Geometrical_representation">Geometrical representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus_knot&action=edit&section=1" title="Edit section: Geometrical representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A torus knot can be rendered geometrically in multiple ways which are <a href="/wiki/Topologically_equivalent" class="mw-redirect" title="Topologically equivalent">topologically equivalent</a> (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following. </p><p>The (<i>p</i>,<i>q</i>)-torus knot winds <i>q</i> times around a circle in the interior of the torus, and <i>p</i> times around its axis of <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a>.<sup id="cite_ref-first_3-0" class="reference"><a href="#cite_note-first-3"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup>. If <i>p</i> and <i>q</i> are not relatively prime, then we have a torus link with more than one component. </p><p>The direction in which the strands of the knot wrap around the torus is also subject to differing conventions. The most common is to have the strands form a right-handed screw for <i>p q > 0</i>.<sup id="cite_ref-livingston_4-0" class="reference"><a href="#cite_note-livingston-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-murasugi_5-0" class="reference"><a href="#cite_note-murasugi-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kawauchi_6-0" class="reference"><a href="#cite_note-kawauchi-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The (<i>p</i>,<i>q</i>)-torus knot can be given by the <a href="/wiki/Parametrization_(geometry)" title="Parametrization (geometry)">parametrization</a> </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 {\begin{aligned}x&=r\cos(p\phi )\\y&=r\sin(p\phi )\\z&=-\sin(q\phi )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=r\cos(p\phi )\\y&=r\sin(p\phi )\\z&=-\sin(q\phi )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08f600c9b23ab648845749fa91b9089709ed140f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.495ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}x&=r\cos(p\phi )\\y&=r\sin(p\phi )\\z&=-\sin(q\phi )\end{aligned}}}"></span></dd></dl> <p>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 r=\cos(q\phi )+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\cos(q\phi )+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd02ff4077928c8e7083374e98bf2aef2376f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.525ex; height:2.843ex;" alt="{\displaystyle r=\cos(q\phi )+2}"></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 0<\phi <2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>ϕ</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\phi <2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c394e57781f81a012c452065aa8b7a81367c489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.239ex; height:2.509ex;" alt="{\displaystyle 0<\phi <2\pi }"></span>. This lies on the surface of the torus given by <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 (r-2)^{2}+z^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r-2)^{2}+z^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed6289068fe08b5dfd6a4dd66d18ab45bb97473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.161ex; height:3.176ex;" alt="{\displaystyle (r-2)^{2}+z^{2}=1}"></span> (in <a href="/wiki/Cylindrical_coordinates" class="mw-redirect" title="Cylindrical coordinates">cylindrical coordinates</a>). </p><p>Other parameterizations are also possible, because knots are defined up to continuous deformation. The illustrations for the (2,3)- and (3,8)-torus knots can be obtained by 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 r=\cos(q\phi )+4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\cos(q\phi )+4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2550551a341e1743da761e611800e88bcc051d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.525ex; height:2.843ex;" alt="{\displaystyle r=\cos(q\phi )+4}"></span>, and in the case of the (2,3)-torus knot by furthermore subtracting respectively <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 3\cos((p-q)\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cos((p-q)\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507d8d3433488199d4f32d6afa827b73d1fb5396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.744ex; height:2.843ex;" alt="{\displaystyle 3\cos((p-q)\phi )}"></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 3\sin((p-q)\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\sin((p-q)\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23b042b4533252c7a0cbf298670bf3e453abf0d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.489ex; height:2.843ex;" alt="{\displaystyle 3\sin((p-q)\phi )}"></span> from the above parameterizations of <i>x</i> and <i>y</i>. The latter generalizes smoothly to any coprime <i>p,q</i> satisfying <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 p<q<2p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo><</mo> <mn>2</mn> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<q<2p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329c05f002a32bc799a7a91fc9ed4244063a6f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.857ex; height:2.509ex;" alt="{\displaystyle p<q<2p}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus_knot&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:TorusKnot-3-8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/TorusKnot-3-8.png" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="200" data-file-height="200" /></a><figcaption>Diagram of a (3,−8)-torus knot.</figcaption></figure> <p>A torus knot is <a href="/wiki/Unknot" title="Unknot">trivial</a> <a href="/wiki/If_and_only_if" title="If and only if">iff</a> either <i>p</i> or <i>q</i> is equal to 1 or −1.<sup id="cite_ref-murasugi_5-1" class="reference"><a href="#cite_note-murasugi-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kawauchi_6-1" class="reference"><a href="#cite_note-kawauchi-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Each nontrivial torus knot is <a href="/wiki/Prime_knot" title="Prime knot">prime</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chiral</a>.<sup id="cite_ref-murasugi_5-2" class="reference"><a href="#cite_note-murasugi-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The (<i>p</i>,<i>q</i>) torus knot is equivalent to the (<i>q</i>,<i>p</i>) torus knot.<sup id="cite_ref-livingston_4-1" class="reference"><a href="#cite_note-livingston-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kawauchi_6-2" class="reference"><a href="#cite_note-kawauchi-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This can be proved by moving the strands on the surface of the torus.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The (<i>p</i>,−<i>q</i>) torus knot is the obverse (mirror image) of the (<i>p</i>,<i>q</i>) torus knot.<sup id="cite_ref-kawauchi_6-3" class="reference"><a href="#cite_note-kawauchi-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The (−<i>p</i>,−<i>q</i>) torus knot is equivalent to the (<i>p</i>,<i>q</i>) torus knot except for the reversed orientation. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:(3,_4)_torus_knot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/%283%2C_4%29_torus_knot.svg/220px-%283%2C_4%29_torus_knot.svg.png" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/%283%2C_4%29_torus_knot.svg/330px-%283%2C_4%29_torus_knot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/%283%2C_4%29_torus_knot.svg/440px-%283%2C_4%29_torus_knot.svg.png 2x" data-file-width="630" data-file-height="396" /></a><figcaption>The (3, 4) torus knot on the unwrapped torus surface, and its braid word</figcaption></figure> <p>Any (<i>p</i>,<i>q</i>)-torus knot can be made from a <a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">closed braid</a> with <i>p</i> strands. The appropriate <a href="/wiki/Braid_word" class="mw-redirect" title="Braid word">braid word</a> is <sup id="cite_ref-lickorish_9-0" class="reference"><a href="#cite_note-lickorish-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </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 (\sigma _{1}\sigma _{2}\cdots \sigma _{p-1})^{q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sigma _{1}\sigma _{2}\cdots \sigma _{p-1})^{q}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3fbfd96308d990e7ea5da3f16b421abdc250fce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.192ex; height:3.009ex;" alt="{\displaystyle (\sigma _{1}\sigma _{2}\cdots \sigma _{p-1})^{q}.}"></span></dd></dl> <p>(This formula assumes the common convention that braid generators are right twists,<sup id="cite_ref-murasugi_5-3" class="reference"><a href="#cite_note-murasugi-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lickorish_9-1" class="reference"><a href="#cite_note-lickorish-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dehornoy_10-0" class="reference"><a href="#cite_note-dehornoy-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-birman_11-0" class="reference"><a href="#cite_note-birman-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> which is not followed by the Wikipedia page on braids.) </p><p>The <a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">crossing number</a> of a (<i>p</i>,<i>q</i>) torus knot with <i>p</i>,<i>q</i> > 0 is given by </p> <dl><dd><i>c</i> = min((<i>p</i>−1)<i>q</i>, (<i>q</i>−1)<i>p</i>).</dd></dl> <p>The <a href="/wiki/Knot_genus" class="mw-redirect" title="Knot genus">genus</a> of a torus knot with <i>p</i>,<i>q</i> > 0 is </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 g={\frac {1}{2}}(p-1)(q-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\frac {1}{2}}(p-1)(q-1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af7770e41051166756b08692fc8052770c097395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.723ex; height:5.176ex;" alt="{\displaystyle g={\frac {1}{2}}(p-1)(q-1).}"></span></dd></dl> <p>The <a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a> of a torus knot is <sup id="cite_ref-livingston_4-2" class="reference"><a href="#cite_note-livingston-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lickorish_9-2" class="reference"><a href="#cite_note-lickorish-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </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 t^{k}{\frac {(t^{pq}-1)(t-1)}{(t^{p}-1)(t^{q}-1)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{k}{\frac {(t^{pq}-1)(t-1)}{(t^{p}-1)(t^{q}-1)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b6e94b15356144f6f408c5190dc66bc37e5e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.763ex; height:6.509ex;" alt="{\displaystyle t^{k}{\frac {(t^{pq}-1)(t-1)}{(t^{p}-1)(t^{q}-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 k=-{\frac {(p-1)(q-1)}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=-{\frac {(p-1)(q-1)}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b07cc81bf7a9458bddc2354c70d177332ea05f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.464ex; height:5.676ex;" alt="{\displaystyle k=-{\frac {(p-1)(q-1)}{2}}.}"></span></dd></dl> <p>The <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a> of a (right-handed) torus knot is given by </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 t^{(p-1)(q-1)/2}{\frac {1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{(p-1)(q-1)/2}{\frac {1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f508733da5fec9f5c9d1e01115e4bf53f440f8f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.087ex; height:6.176ex;" alt="{\displaystyle t^{(p-1)(q-1)/2}{\frac {1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^{2}}}.}"></span></dd></dl> <p>The complement of a torus knot in the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> is a <a href="/wiki/Seifert_fiber_space" title="Seifert fiber space">Seifert-fibered manifold</a>, fibred over the disc with two singular fibres. </p><p>Let <i>Y</i> be the <i>p</i>-fold <a href="/wiki/Dunce_hat_(topology)" title="Dunce hat (topology)">dunce cap</a> with a disk removed from the interior, <i>Z</i> be the <i>q</i>-fold dunce cap with a disk removed from its interior, and <i>X</i> be the quotient space obtained by identifying <i>Y</i> and <i>Z</i> along their boundary circle. The knot complement of the (<i>p</i>, <i>q</i>) -torus knot <a href="/wiki/Deformation_retract" class="mw-redirect" title="Deformation retract">deformation retracts</a> to the space <i>X</i>. Therefore, the <a href="/wiki/Knot_group" title="Knot group">knot group</a> of a torus knot has the <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> </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 \langle x,y\mid x^{p}=y^{q}\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∣</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\mid x^{p}=y^{q}\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76228cfe60384c1af85907dcc05f9ad0febdc036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.549ex; height:2.843ex;" alt="{\displaystyle \langle x,y\mid x^{p}=y^{q}\rangle .}"></span></dd></dl> <p>Torus knots are the only knots whose knot groups have nontrivial <a href="/wiki/Center_of_a_group" class="mw-redirect" title="Center of a group">center</a> (which is infinite cyclic, generated by the element <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^{p}=y^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{p}=y^{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55485de8bc2f98313f72ae94e22d8cd69398eba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.636ex; height:2.676ex;" alt="{\displaystyle x^{p}=y^{q}}"></span> in the presentation above). </p><p>The <a href="/wiki/Stretch_factor" title="Stretch factor">stretch factor</a> of the (<i>p</i>,<i>q</i>) torus knot, as a curve in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, is Ω(min(<i>p</i>,<i>q</i>)), so torus knots have unbounded stretch factors. Undergraduate researcher <a href="/wiki/John_Pardon" title="John Pardon">John Pardon</a> won the 2012 <a href="/wiki/Morgan_Prize" title="Morgan Prize">Morgan Prize</a> for his research proving this result, which solved a problem originally posed by <a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Mikhail Gromov</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Connection_to_complex_hypersurfaces">Connection to complex hypersurfaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus_knot&action=edit&section=3" title="Edit section: Connection to complex hypersurfaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The (<i>p</i>,<i>q</i>)−torus knots arise when considering the link of an isolated complex hypersurface singularity. One intersects the complex hypersurface with a <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">hypersphere</a>, centred at the isolated singular point, and with sufficiently small radius so that it does not enclose, nor encounter, any other singular points. The intersection gives a submanifold of the hypersphere. </p><p>Let <i>p</i> and <i>q</i> be coprime integers, greater than or equal to two. Consider the <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic 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 f:\mathbb {C} ^{2}\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {C} ^{2}\to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d396023b45fcc5f5e0905f6837805d6c88af951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.24ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {C} ^{2}\to \mathbb {C} }"></span> given by <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 f(w,z):=w^{p}+z^{q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(w,z):=w^{p}+z^{q}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ed79d2a88035d6a72f6a033cdc7e4d43e6a44d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.909ex; height:2.843ex;" alt="{\displaystyle f(w,z):=w^{p}+z^{q}.}"></span> 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 V_{f}\subset \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{f}\subset \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0518ed2cbedf1d73adf5bd50c9b37908eabbbe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.322ex; height:3.343ex;" alt="{\displaystyle V_{f}\subset \mathbb {C} ^{2}}"></span> be the set 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 (w,z)\in \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (w,z)\in \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3740a4768de9bc15d6ca7a7f15fde7568319fbc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.168ex; height:3.176ex;" alt="{\displaystyle (w,z)\in \mathbb {C} ^{2}}"></span> such that <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 f(w,z)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(w,z)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ec54ee88e0c0be14af867a73b22aa40690bb35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.782ex; height:2.843ex;" alt="{\displaystyle f(w,z)=0.}"></span> Given a real number <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 0<\varepsilon \ll 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>ε</mi> <mo>≪</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\varepsilon \ll 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5dc5d92f6cf1b71448ba207ff9e88ab3c4c64ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.768ex; height:2.509ex;" alt="{\displaystyle 0<\varepsilon \ll 1,}"></span> we define the real three-sphere <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} _{\varepsilon }^{3}\subset \mathbb {R} ^{4}\hookrightarrow \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">↪</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {R} ^{4}\hookrightarrow \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6485790b44feec760e52be9f0cb7ca25ee4385e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.817ex; height:3.176ex;" alt="{\displaystyle \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {R} ^{4}\hookrightarrow \mathbb {C} ^{2}}"></span> as given by <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 |w|^{2}+|z|^{2}=\varepsilon ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |w|^{2}+|z|^{2}=\varepsilon ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7754002e004df64121bbeec284175a2ba454e3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.172ex; height:3.343ex;" alt="{\displaystyle |w|^{2}+|z|^{2}=\varepsilon ^{2}.}"></span> The 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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> has an isolated <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a> at <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 (0,0)\in \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)\in \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a041d50e79a73a6d931aae9963e13bb0617810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.741ex; height:3.176ex;" alt="{\displaystyle (0,0)\in \mathbb {C} ^{2}}"></span> since <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 \partial f/\partial w=\partial f/\partial z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial f/\partial w=\partial f/\partial z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e766a30e8bf2621ff60e302d0671875e0f7e652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.266ex; height:2.843ex;" alt="{\displaystyle \partial f/\partial w=\partial f/\partial z=0}"></span> if and only if <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 w=z=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mi>z</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=z=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b72a156a26e1a31e82d637725d0e2338f3a00b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.758ex; height:2.176ex;" alt="{\displaystyle w=z=0.}"></span> Thus, we consider the structure 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 V_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc0d1604bbf75069df35d14afa9fac3e883be3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.492ex; height:2.843ex;" alt="{\displaystyle V_{f}}"></span> close 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 (0,0)\in \mathbb {C} ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)\in \mathbb {C} ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14329af20e8fcb070052969f44cf4f56ea156fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.388ex; height:3.176ex;" alt="{\displaystyle (0,0)\in \mathbb {C} ^{2}.}"></span> In order to do this, we consider the intersection <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 V_{f}\cap \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {S} _{\varepsilon }^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>∩</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>⊂</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{f}\cap \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {S} _{\varepsilon }^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/127d47b3c630adf070a98dbd0c392286df17fd3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.513ex; height:3.343ex;" alt="{\displaystyle V_{f}\cap \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {S} _{\varepsilon }^{3}.}"></span> This intersection is the so-called link of the singularity <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 f(w,z)=w^{p}+z^{q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(w,z)=w^{p}+z^{q}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff417b11f1aebc09f185fad50b2e0612129d6c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.262ex; height:2.843ex;" alt="{\displaystyle f(w,z)=w^{p}+z^{q}.}"></span> The link 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 f(w,z)=w^{p}+z^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(w,z)=w^{p}+z^{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f7e8c1ad60d91ecd49b1a734c8965b1c6d453a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.615ex; height:2.843ex;" alt="{\displaystyle f(w,z)=w^{p}+z^{q}}"></span>, where <i>p</i> and <i>q</i> are coprime, and both greater than or equal to two, is exactly the (<i>p</i>,<i>q</i>)−torus knot.<sup id="cite_ref-MILNOR_14-0" class="reference"><a href="#cite_note-MILNOR-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="List">List</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus_knot&action=edit&section=4" title="Edit section: List"><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:Torus_link_(36,3).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Torus_link_%2836%2C3%29.png/220px-Torus_link_%2836%2C3%29.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Torus_link_%2836%2C3%29.png/330px-Torus_link_%2836%2C3%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Torus_link_%2836%2C3%29.png/440px-Torus_link_%2836%2C3%29.png 2x" data-file-width="580" data-file-height="568" /></a><figcaption>(72,4) torus link</figcaption></figure> <ul><li><a href="/wiki/Unknot" title="Unknot">Unknot</a>, <a href="/wiki/Trefoil_knot" title="Trefoil knot">3<sub>1</sub> knot</a> (3,2), <a href="/wiki/Cinquefoil_knot" title="Cinquefoil knot">5<sub>1</sub> knot</a> (5,2), <a href="/wiki/7%E2%82%81_knot" class="mw-redirect" title="7₁ knot">7<sub>1</sub> knot</a> (7,2), 8<sub>19</sub> knot (4,3), <b>9<sub>1</sub> knot</b> (9,2), <b>10<sub>124</sub> knot</b> (5,3)</li></ul> <table class="wikitable sortable" style="text-align:center"> <tbody><tr> <th><a href="/wiki/Knot_tabulation" title="Knot tabulation">Table<br /> #</a> </th> <th><a href="/wiki/Alexander-Briggs_notation" class="mw-redirect" title="Alexander-Briggs notation">A-B</a> </th> <th>Image </th> <th><i>P</i> </th> <th><i>Q</i> </th> <th><a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">Cross<br /> #</a> </th></tr> <tr> <td>0 </td> <td>0<sub>1</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Blue_Unknot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Blue_Unknot.png/60px-Blue_Unknot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Blue_Unknot.png/90px-Blue_Unknot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Blue_Unknot.png/120px-Blue_Unknot.png 2x" data-file-width="1800" data-file-height="1800" /></a></span> </td> <td> </td> <td> </td> <td>0 </td></tr> <tr> <td>3a1 </td> <td>3<sub>1</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:(3-2)_torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/%283-2%29_torus_knot.png/60px-%283-2%29_torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/%283-2%29_torus_knot.png/90px-%283-2%29_torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e4/%283-2%29_torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>2 </td> <td>3 </td> <td>3 </td></tr> <tr> <td>5a2 </td> <td>5<sub>1</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(5,2)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/A_%285%2C2%29-torus_knot.png/60px-A_%285%2C2%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/A_%285%2C2%29-torus_knot.png/90px-A_%285%2C2%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/07/A_%285%2C2%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>2 </td> <td>5 </td> <td>5 </td></tr> <tr> <td>7a7 </td> <td>7<sub>1</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(7,2)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/A_%287%2C2%29-torus_knot.png/60px-A_%287%2C2%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/A_%287%2C2%29-torus_knot.png/90px-A_%287%2C2%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3b/A_%287%2C2%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>2 </td> <td>7 </td> <td>7 </td></tr> <tr> <td>8n3 </td> <td>8<sub>19</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(4,3)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/A_%284%2C3%29-torus_knot.png/60px-A_%284%2C3%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/A_%284%2C3%29-torus_knot.png/90px-A_%284%2C3%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/62/A_%284%2C3%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>3 </td> <td>4 </td> <td>8 </td></tr> <tr> <td>9a41 </td> <td>9<sub>1</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(9,2)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/A_%289%2C2%29-torus_knot.png/60px-A_%289%2C2%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/A_%289%2C2%29-torus_knot.png/90px-A_%289%2C2%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1a/A_%289%2C2%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>2 </td> <td>9 </td> <td>9 </td></tr> <tr> <td>10n21 </td> <td>10<sub>124</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(5,3)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/A_%285%2C3%29-torus_knot.png/60px-A_%285%2C3%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/A_%285%2C3%29-torus_knot.png/90px-A_%285%2C3%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/48/A_%285%2C3%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>3 </td> <td>5 </td> <td>10 </td></tr> <tr> <td>11a367 </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:Simple-knotwork-cross-12crossings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Simple-knotwork-cross-12crossings.svg/60px-Simple-knotwork-cross-12crossings.svg.png" decoding="async" width="60" height="72" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Simple-knotwork-cross-12crossings.svg/90px-Simple-knotwork-cross-12crossings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Simple-knotwork-cross-12crossings.svg/120px-Simple-knotwork-cross-12crossings.svg.png 2x" data-file-width="502" data-file-height="600" /></a></span> </td> <td>2 </td> <td>11 </td> <td>11 </td></tr> <tr> <td>13a4878 </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:(2,13)-Torus_Knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/%282%2C13%29-Torus_Knot.png/60px-%282%2C13%29-Torus_Knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/%282%2C13%29-Torus_Knot.png/90px-%282%2C13%29-Torus_Knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/%282%2C13%29-Torus_Knot.png/120px-%282%2C13%29-Torus_Knot.png 2x" data-file-width="512" data-file-height="512" /></a></span> </td> <td>2 </td> <td>13 </td> <td>13 </td></tr> <tr> <td>14n21881 </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(7,3)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/A_%287%2C3%29-torus_knot.png/60px-A_%287%2C3%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/A_%287%2C3%29-torus_knot.png/90px-A_%287%2C3%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b5/A_%287%2C3%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>3 </td> <td>7 </td> <td>14 </td></tr> <tr> <td>15n41185 </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(5,4)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/A_%285%2C4%29-torus_knot.png/60px-A_%285%2C4%29-torus_knot.png" decoding="async" width="60" height="61" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/A_%285%2C4%29-torus_knot.png/90px-A_%285%2C4%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8a/A_%285%2C4%29-torus_knot.png 2x" data-file-width="99" data-file-height="100" /></a></span> </td> <td>4 </td> <td>5 </td> <td>15 </td></tr> <tr> <td>15a85263 </td> <td> </td> <td> </td> <td>2 </td> <td>15 </td> <td>15 </td></tr> <tr> <td>16n783154 </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(8,3)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/A_%288%2C3%29-torus_knot.png/60px-A_%288%2C3%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/A_%288%2C3%29-torus_knot.png/90px-A_%288%2C3%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c2/A_%288%2C3%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>3 </td> <td>8 </td> <td>16 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>17 </td> <td>17 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>19 </td> <td>19 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>3 </td> <td>10 </td> <td>20 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(7,4)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/A_%287%2C4%29-torus_knot.png/60px-A_%287%2C4%29-torus_knot.png" decoding="async" width="60" height="61" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/A_%287%2C4%29-torus_knot.png/90px-A_%287%2C4%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7c/A_%287%2C4%29-torus_knot.png 2x" data-file-width="99" data-file-height="100" /></a></span> </td> <td>4 </td> <td>7 </td> <td>21 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>21 </td> <td>21 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>3 </td> <td>11 </td> <td>22 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>23 </td> <td>23 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(6,5)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/A_%286%2C5%29-torus_knot.png/60px-A_%286%2C5%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/A_%286%2C5%29-torus_knot.png/90px-A_%286%2C5%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/31/A_%286%2C5%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>5 </td> <td>6 </td> <td>24 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>25 </td> <td>25 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>3 </td> <td>13 </td> <td>26 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(9,4)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/A_%289%2C4%29-torus_knot.png/60px-A_%289%2C4%29-torus_knot.png" decoding="async" width="60" height="61" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/A_%289%2C4%29-torus_knot.png/90px-A_%289%2C4%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/03/A_%289%2C4%29-torus_knot.png 2x" data-file-width="99" data-file-height="100" /></a></span> </td> <td>4 </td> <td>9 </td> <td>27 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>27 </td> <td>27 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(7,5)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/A_%287%2C5%29-torus_knot.png/60px-A_%287%2C5%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/A_%287%2C5%29-torus_knot.png/90px-A_%287%2C5%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c2/A_%287%2C5%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>5 </td> <td>7 </td> <td>28 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>3 </td> <td>14 </td> <td>28 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>29 </td> <td>29 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>31 </td> <td>31 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(8,5)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/A_%288%2C5%29-torus_knot.png/60px-A_%288%2C5%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/A_%288%2C5%29-torus_knot.png/90px-A_%288%2C5%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/de/A_%288%2C5%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>5 </td> <td>8 </td> <td>32 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>3 </td> <td>16 </td> <td>32 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>4 </td> <td>11 </td> <td>33 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>33 </td> <td>33 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>3 </td> <td>17 </td> <td>34 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(7,6)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/A_%287%2C6%29-torus_knot.png/60px-A_%287%2C6%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/A_%287%2C6%29-torus_knot.png/90px-A_%287%2C6%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/ff/A_%287%2C6%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>6 </td> <td>7 </td> <td>35 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>2 </td> <td>35 </td> <td>35 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(9,5)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/A_%289%2C5%29-torus_knot.png/60px-A_%289%2C5%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/A_%289%2C5%29-torus_knot.png/90px-A_%289%2C5%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b1/A_%289%2C5%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>5 </td> <td>9 </td> <td>36 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(8,7)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/A_%288%2C7%29-torus_knot.png/60px-A_%288%2C7%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/A_%288%2C7%29-torus_knot.png/90px-A_%288%2C7%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e0/A_%288%2C7%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>7 </td> <td>8 </td> <td>48 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(9,7)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/A_%289%2C7%29-torus_knot.png/60px-A_%289%2C7%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/A_%289%2C7%29-torus_knot.png/90px-A_%289%2C7%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/2b/A_%289%2C7%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>7 </td> <td>9 </td> <td>54 </td></tr> <tr> <td> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:A_(9,8)-torus_knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/A_%289%2C8%29-torus_knot.png/60px-A_%289%2C8%29-torus_knot.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/A_%289%2C8%29-torus_knot.png/90px-A_%289%2C8%29-torus_knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/88/A_%289%2C8%29-torus_knot.png 2x" data-file-width="99" data-file-height="99" /></a></span> </td> <td>8 </td> <td>9 </td> <td>63 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="g-torus_knot"><i>g</i>-torus knot</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus_knot&action=edit&section=5" title="Edit section: g-torus knot"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>g-torus knot</b> is a closed curve drawn on a <a href="/wiki/Genus_g_surface" title="Genus g surface">g-torus</a>. More technically, it is the homeomorphic image of a circle in <b>S³</b> which can be realized as a subset of a <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> <i>g</i> <a href="/wiki/Handlebody" title="Handlebody">handlebody</a> in <b>S³</b> (whose complement is also a genus <i>g</i> handlebody). If a <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">link</a> is a subset of a genus two handlebody, it is a <b>double torus link</b>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>For genus two, the simplest example of a double torus knot that is not a torus knot is the <a href="/wiki/Figure-eight_knot_(mathematics)" title="Figure-eight knot (mathematics)">figure-eight knot</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torus_knot&action=edit&section=6" title="Edit section: Notes"><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-first-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-first_3-0">^</a></b></span> <span class="reference-text">Note that this use of the roles of p and q is contrary to what appears on.<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> It is also inconsistent with the pictures that appear in: <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></span> </li> </ol></div></div> <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=Torus_knot&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Alternating_knot" title="Alternating knot">Alternating knot</a></li> <li><a href="/wiki/Satellite_knot" title="Satellite knot">Satellite knot</a></li> <li><a href="/wiki/Hyperbolic_knot" class="mw-redirect" title="Hyperbolic knot">Hyperbolic knot</a></li> <li><a href="/wiki/Irrational_winding_of_a_torus" class="mw-redirect" title="Irrational winding of a torus">Irrational winding of a torus</a></li> <li><a href="/wiki/Topopolis" title="Topopolis">Topopolis</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=Torus_knot&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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">Torus Knot on Wolfram Mathworld <a rel="nofollow" class="external autonumber" href="http://mathworld.wolfram.com/TorusKnot.html">[1]</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">"36 Torus Knots", The Knot Atlas. <a rel="nofollow" class="external autonumber" href="http://katlas.math.toronto.edu/wiki/36_Torus_Knots">[2]</a>.</span> </li> <li id="cite_note-livingston-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-livingston_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-livingston_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-livingston_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><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="CITEREFLivingston1993" class="citation book cs1">Livingston, Charles (1993). <i>Knot Theory</i>. Mathematical Association of America. p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88385-027-3" title="Special:BookSources/0-88385-027-3"><bdi>0-88385-027-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knot+Theory&rft.pages=Category%3AWikipedia+articles+needing+page+number+citations+from+November+2017%3Csup+class%3D%22noprint+Inline-Template+%22+style%3D%22white-space%3Anowrap%3B%22%3E%26%2391%3B%3Ci%3EWikipedia%3ACiting+sources%7C%3Cspan+title%3D%22This+citation+requires+a+reference+to+the+specific+page+or+range+of+pages+in+which+the+material+appears.%26%2332%3B%28November+2017%29%22%3Epage-needed%3C%2Fspan%3E%3C%2Fi%3E%26%2393%3B%3C%2Fsup%3E&rft.pub=Mathematical+Association+of+America&rft.date=1993&rft.isbn=0-88385-027-3&rft.aulast=Livingston&rft.aufirst=Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-murasugi-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-murasugi_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-murasugi_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-murasugi_5-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-murasugi_5-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurasugi1996" class="citation book cs1">Murasugi, Kunio (1996). <i>Knot Theory and its Applications</i>. Birkhäuser. p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-7643-3817-2" title="Special:BookSources/3-7643-3817-2"><bdi>3-7643-3817-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knot+Theory+and+its+Applications&rft.pages=Category%3AWikipedia+articles+needing+page+number+citations+from+November+2017%3Csup+class%3D%22noprint+Inline-Template+%22+style%3D%22white-space%3Anowrap%3B%22%3E%26%2391%3B%3Ci%3EWikipedia%3ACiting+sources%7C%3Cspan+title%3D%22This+citation+requires+a+reference+to+the+specific+page+or+range+of+pages+in+which+the+material+appears.%26%2332%3B%28November+2017%29%22%3Epage-needed%3C%2Fspan%3E%3C%2Fi%3E%26%2393%3B%3C%2Fsup%3E&rft.pub=Birkh%C3%A4user&rft.date=1996&rft.isbn=3-7643-3817-2&rft.aulast=Murasugi&rft.aufirst=Kunio&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-kawauchi-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-kawauchi_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-kawauchi_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-kawauchi_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-kawauchi_6-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKawauchi1996" class="citation book cs1">Kawauchi, Akio (1996). <i>A Survey of Knot Theory</i>. Birkhäuser. p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-7643-5124-1" title="Special:BookSources/3-7643-5124-1"><bdi>3-7643-5124-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+Knot+Theory&rft.pages=Category%3AWikipedia+articles+needing+page+number+citations+from+November+2017%3Csup+class%3D%22noprint+Inline-Template+%22+style%3D%22white-space%3Anowrap%3B%22%3E%26%2391%3B%3Ci%3EWikipedia%3ACiting+sources%7C%3Cspan+title%3D%22This+citation+requires+a+reference+to+the+specific+page+or+range+of+pages+in+which+the+material+appears.%26%2332%3B%28November+2017%29%22%3Epage-needed%3C%2Fspan%3E%3C%2Fi%3E%26%2393%3B%3C%2Fsup%3E&rft.pub=Birkh%C3%A4user&rft.date=1996&rft.isbn=3-7643-5124-1&rft.aulast=Kawauchi&rft.aufirst=Akio&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></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="CITEREFNorwood1982" class="citation journal cs1">Norwood, F. H. (1982-01-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1982-0663884-7">"Every two-generator knot is prime"</a>. <i>Proceedings of the American Mathematical Society</i>. <b>86</b> (1): 143–147. <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%2FS0002-9939-1982-0663884-7">10.1090/S0002-9939-1982-0663884-7</a></span>. <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/0002-9939">0002-9939</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/2044414">2044414</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=Every+two-generator+knot+is+prime&rft.volume=86&rft.issue=1&rft.pages=143-147&rft.date=1982-01-01&rft.issn=0002-9939&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2044414%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-1982-0663884-7&rft.aulast=Norwood&rft.aufirst=F.+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9939-1982-0663884-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></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="CITEREFBaker2011" class="citation web cs1">Baker, Kenneth (2011-03-28). <a rel="nofollow" class="external text" href="https://sketchesoftopology.wordpress.com/2011/03/28/pqisqp/">"p q is q p"</a>. <i>Sketches of Topology</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sketches+of+Topology&rft.atitle=p+q+is+q+p&rft.date=2011-03-28&rft.aulast=Baker&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Fsketchesoftopology.wordpress.com%2F2011%2F03%2F28%2Fpqisqp%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-lickorish-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-lickorish_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lickorish_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-lickorish_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLickorish1997" class="citation book cs1">Lickorish, W. B. R. (1997). <i>An Introduction to Knot Theory</i>. Springer. p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98254-X" title="Special:BookSources/0-387-98254-X"><bdi>0-387-98254-X</bdi></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.pages=Category%3AWikipedia+articles+needing+page+number+citations+from+November+2017%3Csup+class%3D%22noprint+Inline-Template+%22+style%3D%22white-space%3Anowrap%3B%22%3E%26%2391%3B%3Ci%3EWikipedia%3ACiting+sources%7C%3Cspan+title%3D%22This+citation+requires+a+reference+to+the+specific+page+or+range+of+pages+in+which+the+material+appears.%26%2332%3B%28November+2017%29%22%3Epage-needed%3C%2Fspan%3E%3C%2Fi%3E%26%2393%3B%3C%2Fsup%3E&rft.pub=Springer&rft.date=1997&rft.isbn=0-387-98254-X&rft.aulast=Lickorish&rft.aufirst=W.+B.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-dehornoy-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-dehornoy_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDehornoyDynnikovRolfsenWiest2000" class="citation book cs1">Dehornoy, P.; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120415032136/http://www.math.unicaen.fr/~dehornoy/Books/Why/Dgr.pdf"><i>Why are Braids Orderable?</i></a> <span class="cs1-format">(PDF)</span>. p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup>. Archived from <a rel="nofollow" class="external text" href="http://www.math.unicaen.fr/~dehornoy/Books/Why/Dgr.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2012-04-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-11-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Why+are+Braids+Orderable%3F&rft.pages=Category%3AWikipedia+articles+needing+page+number+citations+from+November+2017%3Csup+class%3D%22noprint+Inline-Template+%22+style%3D%22white-space%3Anowrap%3B%22%3E%26%2391%3B%3Ci%3EWikipedia%3ACiting+sources%7C%3Cspan+title%3D%22This+citation+requires+a+reference+to+the+specific+page+or+range+of+pages+in+which+the+material+appears.%26%2332%3B%28November+2017%29%22%3Epage-needed%3C%2Fspan%3E%3C%2Fi%3E%26%2393%3B%3C%2Fsup%3E&rft.date=2000&rft.aulast=Dehornoy&rft.aufirst=P.&rft.au=Dynnikov%2C+Ivan&rft.au=Rolfsen%2C+Dale&rft.au=Wiest%2C+Bert&rft_id=http%3A%2F%2Fwww.math.unicaen.fr%2F~dehornoy%2FBooks%2FWhy%2FDgr.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-birman-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-birman_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirmanBrendle2005" class="citation book cs1">Birman, J. S.; <a href="/wiki/Tara_E._Brendle" title="Tara E. Brendle">Brendle, T. E.</a> (2005). "Braids: a Survey". In Menasco, W.; Thistlethwaite, M. (eds.). <i>Handbook of Knot Theory</i>. Elsevier. p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-51452-X" title="Special:BookSources/0-444-51452-X"><bdi>0-444-51452-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Braids%3A+a+Survey&rft.btitle=Handbook+of+Knot+Theory&rft.pages=Category%3AWikipedia+articles+needing+page+number+citations+from+November+2017%3Csup+class%3D%22noprint+Inline-Template+%22+style%3D%22white-space%3Anowrap%3B%22%3E%26%2391%3B%3Ci%3EWikipedia%3ACiting+sources%7C%3Cspan+title%3D%22This+citation+requires+a+reference+to+the+specific+page+or+range+of+pages+in+which+the+material+appears.%26%2332%3B%28November+2017%29%22%3Epage-needed%3C%2Fspan%3E%3C%2Fi%3E%26%2393%3B%3C%2Fsup%3E&rft.pub=Elsevier&rft.date=2005&rft.isbn=0-444-51452-X&rft.aulast=Birman&rft.aufirst=J.+S.&rft.au=Brendle%2C+T.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKehoe2012" class="citation cs2">Kehoe, Elaine (April 2012), "2012 Morgan Prize", <i><a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a></i>, vol. 59, no. 4, pp. 569–571, <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%2Fnoti825">10.1090/noti825</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=2012+Morgan+Prize&rft.volume=59&rft.issue=4&rft.pages=569-571&rft.date=2012-04&rft_id=info%3Adoi%2F10.1090%2Fnoti825&rft.aulast=Kehoe&rft.aufirst=Elaine&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPardon2011" class="citation cs2">Pardon, John (2011), "On the distortion of knots on embedded surfaces", <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, Second Series, <b>174</b> (1): 637–646, <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/1010.1972">1010.1972</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/10.4007%2Fannals.2011.174.1.21">10.4007/annals.2011.174.1.21</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=2811613">2811613</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:55567836">55567836</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=On+the+distortion+of+knots+on+embedded+surfaces&rft.volume=174&rft.issue=1&rft.pages=637-646&rft.date=2011&rft_id=info%3Aarxiv%2F1010.1972&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2811613%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A55567836%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4007%2Fannals.2011.174.1.21&rft.aulast=Pardon&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-MILNOR-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-MILNOR_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilnor1968" class="citation book cs1">Milnor, J. (1968). <i>Singular Points of Complex Hypersurfaces</i>. Princeton University Press. p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08065-8" title="Special:BookSources/0-691-08065-8"><bdi>0-691-08065-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Singular+Points+of+Complex+Hypersurfaces&rft.pages=Category%3AWikipedia+articles+needing+page+number+citations+from+November+2017%3Csup+class%3D%22noprint+Inline-Template+%22+style%3D%22white-space%3Anowrap%3B%22%3E%26%2391%3B%3Ci%3EWikipedia%3ACiting+sources%7C%3Cspan+title%3D%22This+citation+requires+a+reference+to+the+specific+page+or+range+of+pages+in+which+the+material+appears.%26%2332%3B%28November+2017%29%22%3Epage-needed%3C%2Fspan%3E%3C%2Fi%3E%26%2393%3B%3C%2Fsup%3E&rft.pub=Princeton+University+Press&rft.date=1968&rft.isbn=0-691-08065-8&rft.aulast=Milnor&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRolfsen1976" class="citation book cs1">Rolfsen, Dale (1976). <i>Knots and Links</i>. Publish or Perish, Inc. p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-914098-16-0" title="Special:BookSources/0-914098-16-0"><bdi>0-914098-16-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots+and+Links&rft.pages=Category%3AWikipedia+articles+needing+page+number+citations+from+November+2017%3Csup+class%3D%22noprint+Inline-Template+%22+style%3D%22white-space%3Anowrap%3B%22%3E%26%2391%3B%3Ci%3EWikipedia%3ACiting+sources%7C%3Cspan+title%3D%22This+citation+requires+a+reference+to+the+specific+page+or+range+of+pages+in+which+the+material+appears.%26%2332%3B%28November+2017%29%22%3Epage-needed%3C%2Fspan%3E%3C%2Fi%3E%26%2393%3B%3C%2Fsup%3E&rft.pub=Publish+or+Perish%2C+Inc.&rft.date=1976&rft.isbn=0-914098-16-0&rft.aulast=Rolfsen&rft.aufirst=Dale&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHill1999" class="citation journal cs1">Hill, Peter (December 1999). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/doi/abs/10.1142/S0218216599000651">"On Double-Torus Knots (I)"</a>. <i>Journal of Knot Theory and Its Ramifications</i>. <b>08</b> (8): 1009–1048. <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%2FS0218216599000651">10.1142/S0218216599000651</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/0218-2165">0218-2165</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=On+Double-Torus+Knots+%28I%29&rft.volume=08&rft.issue=8&rft.pages=1009-1048&rft.date=1999-12&rft_id=info%3Adoi%2F10.1142%2FS0218216599000651&rft.issn=0218-2165&rft.aulast=Hill&rft.aufirst=Peter&rft_id=https%3A%2F%2Fwww.worldscientific.com%2Fdoi%2Fabs%2F10.1142%2FS0218216599000651&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNorwood1989" class="citation journal cs1">Norwood, Frederick (November 1989). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0166-8641%2889%2990105-3">"Curves on surfaces"</a>. <i>Topology and Its Applications</i>. <b>33</b> (3): 241–246. <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%2F0166-8641%2889%2990105-3">10.1016/0166-8641(89)90105-3</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+and+Its+Applications&rft.atitle=Curves+on+surfaces&rft.volume=33&rft.issue=3&rft.pages=241-246&rft.date=1989-11&rft_id=info%3Adoi%2F10.1016%2F0166-8641%2889%2990105-3&rft.aulast=Norwood&rft.aufirst=Frederick&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0166-8641%252889%252990105-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span> </li> </ol></div></div> <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=Torus_knot&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>"<a rel="nofollow" class="external text" href="https://katlas.org/wiki/36_Torus_Knots">36 Torus Knots</a>", <i><a href="/wiki/The_Knot_Atlas" title="The Knot Atlas">The Knot Atlas</a></i>.</li> <li><span class="citation mathworld" id="Reference-Mathworld-Torus_Knot"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TorusKnot.html">"Torus Knot"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</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=Torus+Knot&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTorusKnot.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorus+knot" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.flexr.org/blog/?p=135">Torus knot renderer in Actionscript</a></li> <li><a rel="nofollow" class="external text" href="http://www.blackpawn.com/texts/pqtorus/">Fun with the PQ-Torus Knot</a></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 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<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 class="mw-selflink selflink">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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