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typeof="mw:File/Thumb"><a href="/wiki/File:Pretzel_knot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Pretzel_knot.svg/140px-Pretzel_knot.svg.png" decoding="async" width="140" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Pretzel_knot.svg/210px-Pretzel_knot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Pretzel_knot.svg/280px-Pretzel_knot.svg.png 2x" data-file-width="283" data-file-height="312" /></a><figcaption>The <a href="/wiki/(%E2%88%922,3,7)_pretzel_knot" title="(−2,3,7) pretzel knot">(−2,3,7) pretzel knot</a> has two right-handed twists in its first <b>tangle</b>, three left-handed twists in its second, and seven left-handed twists in its third.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>tangle</b> is generally one of two related concepts: </p> <ul><li>In <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Conway's</a> definition, an <b><i>n</i>-tangle</b> is a proper <a href="/wiki/Embedding" title="Embedding">embedding</a> of the disjoint union of <i>n</i> arcs into a <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">3-ball</a>; the embedding must send the endpoints of the arcs to 2<i>n</i> marked points on the ball's boundary.</li> <li>In <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">link theory</a>, a tangle is an embedding of <i>n</i> arcs and <i>m</i> circles into <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 \mathbf {R} ^{2}\times [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{2}\times [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b543333db3f8648a3d346a70d258e7af2de16c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.55ex; height:3.176ex;" alt="{\displaystyle \mathbf {R} ^{2}\times [0,1]}"></span> – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance.</li></ul> <p>(A quite different use of 'tangle' appears in Graph minors X. Obstructions to tree-decomposition by N. Robertson and P. D. Seymour, <i><a href="/wiki/Journal_of_Combinatorial_Theory" title="Journal of Combinatorial Theory">Journal of Combinatorial Theory</a></i> B 52 (1991) 153–190, who used it to describe separation in graphs. This usage has been extended to <a href="/wiki/Matroids" class="mw-redirect" title="Matroids">matroids</a>.) </p><p>The balance of this article discusses Conway's sense of tangles; for the link theory sense, see <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">that article</a>. </p><p>Two <i>n</i>-tangles are considered equivalent if there is an <a href="/wiki/Ambient_isotopy" title="Ambient isotopy">ambient isotopy</a> of one tangle to the other keeping the boundary of the 3-ball fixed. <b>Tangle theory</b> can be considered analogous to <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a> except, instead of closed loops, strings whose ends are nailed down are used. See also <a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">braid theory</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Tangle_diagrams">Tangle diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangle_(mathematics)&action=edit&section=1" title="Edit section: Tangle diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Without loss of generality, consider the marked points on the 3-ball boundary to lie on a great circle. The tangle can be arranged to be in <a href="/wiki/General_position" title="General position">general position</a> with respect to the projection onto the flat disc bounded by the great circle. The projection then gives us a <b>tangle diagram</b>, where we make note of over and undercrossings as with <a href="/wiki/Knot_diagram" class="mw-redirect" title="Knot diagram">knot diagrams</a>. </p><p>Tangles often show up as tangle diagrams in knot or link diagrams and can be used as building blocks for <a href="/wiki/Link_diagram" class="mw-redirect" title="Link diagram">link diagrams</a>, e.g. <a href="/wiki/Pretzel_link" title="Pretzel link">pretzel links</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Rational_and_algebraic_tangles">Rational and algebraic tangles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangle_(mathematics)&action=edit&section=2" title="Edit section: Rational and algebraic tangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Tangle_Operations.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Tangle_Operations.svg/300px-Tangle_Operations.svg.png" decoding="async" width="300" height="286" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Tangle_Operations.svg/450px-Tangle_Operations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Tangle_Operations.svg/600px-Tangle_Operations.svg.png 2x" data-file-width="840" data-file-height="800" /></a><figcaption><div style="text-align:center"><b>Some operations on tangles:</b></div> <b>Left:</b> A tangle <i>a</i> and its reflection <i><sup>−</sup>a</i>. <b>Top right:</b> Tangle addition, denoted by <i>a + b</i>. <b>Center right:</b> Tangle product, denoted by <i>a b</i>, equivalent to <i><sup>−</sup>a + b</i>. <b>Bottom right:</b> Ramification, denoted by <i>a , b</i>, equivalent to <i><sup>−</sup>a + <sup>−</sup>b</i></figcaption></figure> <p>A <b>rational tangle</b> is a 2-tangle that is homeomorphic to the trivial 2-tangle by a map of pairs consisting of the 3-ball and two arcs. The four endpoints of the arcs on the boundary circle of a tangle diagram are usually referred as NE, NW, SW, SE, with the symbols referring to the compass directions. </p><p>An arbitrary tangle diagram of a rational tangle may look very complicated, but there is always a diagram of a particular simple form: start with a tangle diagram consisting of two horizontal (vertical) arcs; add a "twist", i.e. a single crossing by switching the NE and SE endpoints (SW and SE endpoints); continue by adding more twists using either the NE and SE endpoints or the SW and SE endpoints. One can suppose each twist does not change the diagram inside a disc containing previously created crossings. </p><p>We can describe such a diagram by considering the numbers given by consecutive twists around the same set of endpoints, e.g. (2, 1, -3) means start with two horizontal arcs, then 2 twists using NE/SE endpoints, then 1 twist using SW/SE endpoints, and then 3 twists using NE/SE endpoints but twisting in the opposite direction from before. The list begins with 0 if you start with two vertical arcs. The diagram with two horizontal arcs is then (0), but we assign (0, 0) to the diagram with vertical arcs. A convention is needed to describe a "positive" or "negative" twist. Often, "rational tangle" refers to a list of numbers representing a simple diagram as described. </p><p>The <b>fraction</b> of a rational tangle <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 (a_{0},a_{1},a_{2},\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0},a_{1},a_{2},\dots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b1ae74553708bab8db29b5757a327bd94938bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.487ex; height:2.843ex;" alt="{\displaystyle (a_{0},a_{1},a_{2},\dots )}"></span> is then defined as the number given by the continued fraction <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 [a_{n},a_{n-1},a_{n-2},\dots ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a_{n},a_{n-1},a_{n-2},\dots ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6bcac6d680129ee8620ccf78b7e7c20f76fb020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.665ex; height:2.843ex;" alt="{\displaystyle [a_{n},a_{n-1},a_{n-2},\dots ]}"></span>. The fraction given by (0,0) is defined as <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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>. Conway proved that the fraction is well-defined and completely determines the rational tangle up to tangle equivalence.<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> An accessible proof of this fact is given 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> Conway also defined a fraction of an arbitrary tangle by using the <a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Operations_on_tangles">Operations on tangles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangle_(mathematics)&action=edit&section=3" title="Edit section: Operations on tangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is an "arithmetic" of tangles with addition, multiplication, and reciprocal operations. An algebraic tangle is obtained from the addition and multiplication of rational tangles. </p><p>The <b>numerator closure</b> of a rational tangle is defined as the link obtained by joining the "north" endpoints together and the "south" endpoints also together. The <b>denominator closure</b> is defined similarly by grouping the "east" and "west" endpoints. <a href="/wiki/Rational_link" class="mw-redirect" title="Rational link">Rational links</a> are defined to be such closures of rational tangles. </p> <div class="mw-heading mw-heading2"><h2 id="Conway_notation">Conway notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangle_(mathematics)&action=edit&section=4" title="Edit section: Conway notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation (knot theory)</a></div> <p>One motivation for Conway's study of tangles was to provide a notation for knots more systematic than the traditional enumeration found in tables. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangle_(mathematics)&action=edit&section=5" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tangles have been shown to be useful in studying <a href="/wiki/DNA_topology" class="mw-redirect" title="DNA topology">DNA topology</a>. The action of a given <a href="/wiki/Enzyme" title="Enzyme">enzyme</a> can be analysed with the help of tangle theory.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangle_(mathematics)&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Disentanglement_puzzle" title="Disentanglement puzzle">Tanglement puzzle</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=Tangle_(mathematics)&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFConway1970" class="citation book cs1"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, J. H.</a> (1970). <a rel="nofollow" class="external text" href="http://www.maths.ed.ac.uk/~aar/papers/conway.pdf">"An Enumeration of Knots and Links, and Some of Their Algebraic Properties"</a> <span class="cs1-format">(PDF)</span>. In Leech, J. (ed.). <i>Computational Problems in Abstract Algebra</i>. Oxford, England: Pergamon Press. pp. <span class="nowrap">329–</span>358.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+Enumeration+of+Knots+and+Links%2C+and+Some+of+Their+Algebraic+Properties&rft.btitle=Computational+Problems+in+Abstract+Algebra&rft.place=Oxford%2C+England&rft.pages=%3Cspan+class%3D%22nowrap%22%3E329-%3C%2Fspan%3E358&rft.pub=Pergamon+Press&rft.date=1970&rft.aulast=Conway&rft.aufirst=J.+H.&rft_id=http%3A%2F%2Fwww.maths.ed.ac.uk%2F~aar%2Fpapers%2Fconway.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangle+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKauffmanLambropoulou2004" class="citation journal cs1"><a href="/wiki/Louis_Kauffman" title="Louis Kauffman">Kauffman, Louis H.</a>; Lambropoulou, Sofia (12 Jan 2004). "On the classification of rational tangles". <i><a href="/wiki/Advances_in_Applied_Mathematics" title="Advances in Applied Mathematics">Advances in Applied Mathematics</a></i>. <b>33</b> (2): <span class="nowrap">199–</span>237. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0311499">math/0311499</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003math.....11499K">2003math.....11499K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aam.2003.06.002">10.1016/j.aam.2003.06.002</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:119143716">119143716</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Applied+Mathematics&rft.atitle=On+the+classification+of+rational+tangles&rft.volume=33&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E199-%3C%2Fspan%3E237&rft.date=2004-01-12&rft_id=info%3Aarxiv%2Fmath%2F0311499&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119143716%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.aam.2003.06.002&rft_id=info%3Abibcode%2F2003math.....11499K&rft.aulast=Kauffman&rft.aufirst=Louis+H.&rft.au=Lambropoulou%2C+Sofia&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangle+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErnstSumners1990" class="citation journal cs1">Ernst, C.; Sumners, D. W. (November 1990). "A calculus for rational tangles: applications to DNA recombination". <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. <b>108</b> (3): <span class="nowrap">489–</span>515. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1990MPCPS.108..489E">1990MPCPS.108..489E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0305004100069383">10.1017/s0305004100069383</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/0305-0041">0305-0041</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.atitle=A+calculus+for+rational+tangles%3A+applications+to+DNA+recombination&rft.volume=108&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E489-%3C%2Fspan%3E515&rft.date=1990-11&rft.issn=0305-0041&rft_id=info%3Adoi%2F10.1017%2Fs0305004100069383&rft_id=info%3Abibcode%2F1990MPCPS.108..489E&rft.aulast=Ernst&rft.aufirst=C.&rft.au=Sumners%2C+D.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangle+%28mathematics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangle_(mathematics)&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdams2004" class="citation book cs1">Adams, C. C. (2004). <i>The Knot Book: An elementary introduction to the mathematical theory of knots</i>. Providence, RI: American Mathematical Society. pp. xiv+307. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-3678-1" title="Special:BookSources/0-8218-3678-1"><bdi>0-8218-3678-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Knot+Book%3A+An+elementary+introduction+to+the+mathematical+theory+of+knots&rft.place=Providence%2C+RI&rft.pages=xiv%2B307&rft.pub=American+Mathematical+Society&rft.date=2004&rft.isbn=0-8218-3678-1&rft.aulast=Adams&rft.aufirst=C.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangle+%28mathematics%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangle_(mathematics)&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMacKay" class="citation web cs1"><a href="/wiki/David_J._C._MacKay" title="David J. C. MacKay">MacKay, David</a>. <a rel="nofollow" class="external text" href="http://www.inference.org.uk/mackay/metapost">"Metapost code for drawing tangles and other pictures"</a>. <i>Inference Group</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-04-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Inference+Group&rft.atitle=Metapost+code+for+drawing+tangles+and+other+pictures&rft.aulast=MacKay&rft.aufirst=David&rft_id=http%3A%2F%2Fwww.inference.org.uk%2Fmackay%2Fmetapost&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangle+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldmanKauffman1997" class="citation journal cs1">Goldman, Jay R.; <a href="/wiki/Louis_Kauffman" title="Louis Kauffman">Kauffman, Louis H.</a> (1997). <a rel="nofollow" class="external text" href="http://www.math.uic.edu/~kauffman/RTang.pdf">"Rational Tangles"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Advances_in_Applied_Mathematics" title="Advances in Applied Mathematics">Advances in Applied Mathematics</a></i>. <b>18</b> (3): <span class="nowrap">300–</span>332. <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/10.1006%2Faama.1996.0511">10.1006/aama.1996.0511</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Applied+Mathematics&rft.atitle=Rational+Tangles&rft.volume=18&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E300-%3C%2Fspan%3E332&rft.date=1997&rft_id=info%3Adoi%2F10.1006%2Faama.1996.0511&rft.aulast=Goldman&rft.aufirst=Jay+R.&rft.au=Kauffman%2C+Louis+H.&rft_id=http%3A%2F%2Fwww.math.uic.edu%2F~kauffman%2FRTang.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangle+%28mathematics%29" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Tangle_(mathematics)&oldid=1218185010">https://en.wikipedia.org/w/index.php?title=Tangle_(mathematics)&oldid=1218185010</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Knot_theory" title="Category:Knot theory">Knot theory</a></li><li><a href="/wiki/Category:John_Horton_Conway" title="Category:John Horton Conway">John Horton Conway</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden category: <ul><li><a href="/wiki/Category:Wikipedia_articles_needing_reorganization_from_April_2024" title="Category:Wikipedia articles needing reorganization from April 2024">Wikipedia articles needing reorganization from April 2024</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 April 2024, at 06:59<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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