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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Unique knot with a crossing number of four</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">This article is about the mathematical concept. For the knot, see <a href="/wiki/Figure-eight_knot" title="Figure-eight knot">Figure-eight knot</a>. For other uses, see <a href="/wiki/Figure_8_(disambiguation)" class="mw-redirect mw-disambig" title="Figure 8 (disambiguation)">Figure 8</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox" style="width:16em;"><tbody><tr><th colspan="2" class="infobox-above" style="background:#ffff99">Figure-eight knot</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Blue_Figure-Eight_Knot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Blue_Figure-Eight_Knot.png/220px-Blue_Figure-Eight_Knot.png" decoding="async" width="220" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Blue_Figure-Eight_Knot.png/330px-Blue_Figure-Eight_Knot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Blue_Figure-Eight_Knot.png/440px-Blue_Figure-Eight_Knot.png 2x" data-file-width="1584" data-file-height="1800" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Common name</th><td class="infobox-data"><a href="/wiki/Figure-eight_knot" title="Figure-eight knot">Figure-eight knot</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Arf_invariant_of_a_knot" title="Arf invariant of a knot">Arf invariant</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Braid_length" class="mw-redirect" title="Braid length">Braid length</a></th><td class="infobox-data">4</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Braid_number" class="mw-redirect" title="Braid number">Braid no.</a></th><td class="infobox-data">3</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Bridge_number" title="Bridge number">Bridge no.</a></th><td class="infobox-data">2</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Crosscap_number" title="Crosscap number">Crosscap no.</a></th><td class="infobox-data">2</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">Crossing no.</a></th><td class="infobox-data">4</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Knot_genus" class="mw-redirect" title="Knot genus">Genus</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Hyperbolic_volume" title="Hyperbolic volume">Hyperbolic volume</a></th><td class="infobox-data">2.02988</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Stick_number" title="Stick number">Stick no.</a></th><td class="infobox-data">7</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Unknotting_number" title="Unknotting number">Unknotting no.</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation</a></th><td class="infobox-data">[22]</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Knot_theory#Alexander–Briggs_notation" title="Knot theory">A–B notation</a></th><td class="infobox-data">4<sub>1</sub></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dowker_notation" class="mw-redirect" title="Dowker notation">Dowker notation</a></th><td class="infobox-data">4, 6, 8, 2</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_mathematical_knots_and_links" title="List of mathematical knots and links">Last /<span class="nowrap"> </span>Next</a></th><td class="infobox-data"><a href="/wiki/3_1_knot" class="mw-redirect" title="3 1 knot">3<sub>1</sub></a> / <a href="/wiki/5_1_knot" class="mw-redirect" title="5 1 knot">5<sub>1</sub></a></td></tr><tr><th colspan="2" class="infobox-header">Other</th></tr><tr><td colspan="2" class="infobox-full-data"><a href="/wiki/Alternating_knot" title="Alternating knot">alternating</a>, <a href="/wiki/Hyperbolic_knot" class="mw-redirect" title="Hyperbolic knot">hyperbolic</a>, <a href="/wiki/Fibered_knot" title="Fibered knot">fibered</a>, <a href="/wiki/Prime_knot" title="Prime knot">prime</a>, <a href="/wiki/Fully_amphichiral_knot" class="mw-redirect" title="Fully amphichiral knot">fully amphichiral</a>, <a href="/wiki/Twist_knot" title="Twist knot">twist</a></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Figure8knot-mathematical-knot-theory.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Figure8knot-mathematical-knot-theory.svg/100px-Figure8knot-mathematical-knot-theory.svg.png" decoding="async" width="100" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Figure8knot-mathematical-knot-theory.svg/150px-Figure8knot-mathematical-knot-theory.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Figure8knot-mathematical-knot-theory.svg/200px-Figure8knot-mathematical-knot-theory.svg.png 2x" data-file-width="453" data-file-height="939" /></a><figcaption>Figure-eight knot of practical knot-tying, with ends joined</figcaption></figure> <p>In <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>, a <b>figure-eight knot</b> (also called <b>Listing's knot</b><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>) is the unique knot with a <a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">crossing number</a> of four. This makes it the knot with the third-smallest possible crossing number, after the <a href="/wiki/Unknot" title="Unknot">unknot</a> and the <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knot</a>. The figure-eight knot is a <a href="/wiki/Prime_knot" title="Prime knot">prime knot</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Origin_of_name">Origin of name</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figure-eight_knot_(mathematics)&action=edit&section=1" title="Edit section: Origin of name"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The name is given because tying a normal <a href="/wiki/Figure-eight_knot" title="Figure-eight knot">figure-eight knot</a> in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot. </p> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figure-eight_knot_(mathematics)&action=edit&section=2" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple parametric representation of the figure-eight knot is as the set of all points (<i>x</i>,<i>y</i>,<i>z</i>) where </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&=\left(2+\cos {(2t)}\right)\cos {(3t)}\\y&=\left(2+\cos {(2t)}\right)\sin {(3t)}\\z&=\sin {(4t)}\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> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=\left(2+\cos {(2t)}\right)\cos {(3t)}\\y&=\left(2+\cos {(2t)}\right)\sin {(3t)}\\z&=\sin {(4t)}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c458b7a8910eb4441a525d27b4e7a43acdb4a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.998ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}x&=\left(2+\cos {(2t)}\right)\cos {(3t)}\\y&=\left(2+\cos {(2t)}\right)\sin {(3t)}\\z&=\sin {(4t)}\end{aligned}}}"></span></dd></dl> <p>for <i>t</i> varying over the real numbers (see 2D visual realization at bottom right). </p><p>The figure-eight knot is <a href="/wiki/Prime_knot" title="Prime knot">prime</a>, <a href="/wiki/Alternating_knot" title="Alternating knot">alternating</a>, <a href="/wiki/Rational_knot" class="mw-redirect" title="Rational knot">rational</a> with an associated value of 5/3,<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> and is <a href="/wiki/Chiral_knot" title="Chiral knot">achiral</a>. The figure-eight knot is also a <a href="/wiki/Fibered_knot" title="Fibered knot">fibered knot</a>. This follows from other, less simple (but very interesting) representations of the knot: </p><p>(1) It is a <i>homogeneous</i><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Closed_braid" class="mw-redirect" title="Closed braid">closed braid</a> (namely, the closure of the 3-string braid σ<sub>1</sub>σ<sub>2</sub><sup>−1</sup>σ<sub>1</sub>σ<sub>2</sub><sup>−1</sup>), and a theorem of <a href="/wiki/John_Stallings" class="mw-redirect" title="John Stallings">John Stallings</a> shows that any closed homogeneous braid is fibered. </p><p>(2) It is the link at (0,0,0,0) of an <a href="/w/index.php?title=Isolated_critical_point&action=edit&redlink=1" class="new" title="Isolated critical point (page does not exist)">isolated critical point</a> of a real-polynomial map <var>F</var>: <b>R</b><sup>4</sup>→<b>R</b><sup>2</sup>, so (according to a theorem of <a href="/wiki/John_Milnor" title="John Milnor">John Milnor</a>) the <a href="/wiki/Milnor_map" title="Milnor map">Milnor map</a> of <var>F</var> is actually a fibration. <a href="/w/index.php?title=Bernard_Perron&action=edit&redlink=1" class="new" title="Bernard Perron (page does not exist)">Bernard Perron</a> found the first such <var>F</var> for this knot, namely, </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 F(x,y,z,t)=G(x,y,z^{2}-t^{2},2zt),\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mn>2</mn> <mi>z</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y,z,t)=G(x,y,z^{2}-t^{2},2zt),\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc4a34692d9630816377f1421929bd3e5255f20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:34.389ex; height:3.176ex;" alt="{\displaystyle F(x,y,z,t)=G(x,y,z^{2}-t^{2},2zt),\,\!}"></span></dd></dl> <p>where </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}G(x,y,z,t)=\ &(z(x^{2}+y^{2}+z^{2}+t^{2})+x(6x^{2}-2y^{2}-2z^{2}-2t^{2}),\\&\ tx{\sqrt {2}}+y(6x^{2}-2y^{2}-2z^{2}-2t^{2})).\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>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <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> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mi>t</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G(x,y,z,t)=\ &(z(x^{2}+y^{2}+z^{2}+t^{2})+x(6x^{2}-2y^{2}-2z^{2}-2t^{2}),\\&\ tx{\sqrt {2}}+y(6x^{2}-2y^{2}-2z^{2}-2t^{2})).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd2863b966e0bfe99a8a32a7e6face21bcbcc3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:64.976ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}G(x,y,z,t)=\ &(z(x^{2}+y^{2}+z^{2}+t^{2})+x(6x^{2}-2y^{2}-2z^{2}-2t^{2}),\\&\ tx{\sqrt {2}}+y(6x^{2}-2y^{2}-2z^{2}-2t^{2})).\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Mathematical_properties">Mathematical properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figure-eight_knot_(mathematics)&action=edit&section=3" title="Edit section: Mathematical properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The figure-eight knot has played an important role historically (and continues to do so) in the theory of <a href="/wiki/3-manifold" title="3-manifold">3-manifolds</a>. Sometime in the mid-to-late 1970s, <a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a> showed that the figure-eight was <a href="/wiki/Hyperbolic_knot" class="mw-redirect" title="Hyperbolic knot">hyperbolic</a>, by <a href="/wiki/Manifold_decomposition" title="Manifold decomposition">decomposing</a> its <a href="/wiki/Knot_complement" title="Knot complement">complement</a> into two <a href="/wiki/Ideal_point" title="Ideal point">ideal</a> <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic</a> <a href="/wiki/Tetrahedra" class="mw-redirect" title="Tetrahedra">tetrahedra</a>. (Robert Riley and Troels Jørgensen, working independently of each other, had earlier shown that the figure-eight knot was hyperbolic by other means.) This construction, new at the time, led him to many powerful results and methods. For example, he was able to show that all but ten <a href="/wiki/Dehn_surgery" title="Dehn surgery">Dehn surgeries</a> on the figure-eight knot resulted in non-<a href="/wiki/Haken_manifold" title="Haken manifold">Haken</a>, non-<a href="/wiki/Seifert_fiber_space" title="Seifert fiber space">Seifert-fibered</a> <a href="/wiki/Prime_decomposition_(3-manifold)" class="mw-redirect" title="Prime decomposition (3-manifold)">irreducible</a> 3-manifolds; these were the first such examples. Many more have been discovered by generalizing Thurston's construction to other knots and links. </p><p>The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible <a href="/wiki/Hyperbolic_volume_(knot)" class="mw-redirect" title="Hyperbolic volume (knot)">volume</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 6\Lambda (\pi /3)\approx 2.02988...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>2.02988...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6\Lambda (\pi /3)\approx 2.02988...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090e47620a94f33b34aa3b8d09ea2eb01741a09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.902ex; height:2.843ex;" alt="{\displaystyle 6\Lambda (\pi /3)\approx 2.02988...}"></span> (sequence <span class="nowrap external"><a href="//oeis.org/A091518" class="extiw" title="oeis:A091518">A091518</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), 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 \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> is the <a href="/wiki/Lobachevsky_function" class="mw-redirect" title="Lobachevsky function">Lobachevsky function</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a <a href="/wiki/Covering_space" title="Covering space">double-cover</a> of the <a href="/wiki/Gieseking_manifold" title="Gieseking manifold">Gieseking manifold</a>, which has the smallest volume among non-compact hyperbolic 3-manifolds. </p><p>The figure-eight knot and the <a href="/wiki/(%E2%88%922,3,7)_pretzel_knot" title="(−2,3,7) pretzel knot">(−2,3,7) pretzel knot</a> are the only two hyperbolic knots known to have more than 6 <i>exceptional surgeries</i>, Dehn surgeries resulting in a non-hyperbolic 3-manifold; they have 10 and 7, respectively. A theorem of <a href="/wiki/Marc_Lackenby" title="Marc Lackenby">Lackenby</a> and Meyerhoff, whose proof relies on the <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">geometrization conjecture</a> and <a href="/wiki/Computer-assisted_proof" title="Computer-assisted proof">computer assistance</a>, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot. However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound (except for the two knots mentioned) is 6. </p> <table> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Figure8knot-math-square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Figure8knot-math-square.svg/220px-Figure8knot-math-square.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Figure8knot-math-square.svg/330px-Figure8knot-math-square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Figure8knot-math-square.svg/440px-Figure8knot-math-square.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Simple squared depiction of figure-eight configuration.</figcaption></figure></td> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Figure8knot-rose-limacon-curve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Figure8knot-rose-limacon-curve.svg/220px-Figure8knot-rose-limacon-curve.svg.png" decoding="async" width="220" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Figure8knot-rose-limacon-curve.svg/330px-Figure8knot-rose-limacon-curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Figure8knot-rose-limacon-curve.svg/440px-Figure8knot-rose-limacon-curve.svg.png 2x" data-file-width="696" data-file-height="600" /></a><figcaption>Symmetric depiction generated by parametric equations.</figcaption></figure></td> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Superf%C3%ADcie_-_bordo_n%C3%B3_figura-oito.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Superf%C3%ADcie_-_bordo_n%C3%B3_figura-oito.jpg/220px-Superf%C3%ADcie_-_bordo_n%C3%B3_figura-oito.jpg" decoding="async" width="220" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Superf%C3%ADcie_-_bordo_n%C3%B3_figura-oito.jpg/330px-Superf%C3%ADcie_-_bordo_n%C3%B3_figura-oito.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Superf%C3%ADcie_-_bordo_n%C3%B3_figura-oito.jpg/440px-Superf%C3%ADcie_-_bordo_n%C3%B3_figura-oito.jpg 2x" data-file-width="3787" data-file-height="3169" /></a><figcaption>Mathematical surface Illustrating Figure-eight knot</figcaption></figure> </td></tr></tbody></table> <p>The figure-eight knot has genus 1 and is fibered. Therefore its complement fibers over the circle, the fibers being <a href="/wiki/Seifert_surface" title="Seifert surface">Seifert surfaces</a> which are 2-dimensional tori with one boundary component. The <a href="/wiki/Fibered_knot" title="Fibered knot">monodromy map</a> is then a homeomorphism of the 2-torus, which can be represented in this case by the matrix <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{smallmatrix}2&1\\1&1\end{smallmatrix}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\begin{smallmatrix}2&1\\1&1\end{smallmatrix}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b42752e3b3f4aeba16de7eb56e3776d3a5e3b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.919ex; margin-bottom: -0.253ex; width:4.978ex; height:3.176ex;" alt="{\displaystyle ({\begin{smallmatrix}2&1\\1&1\end{smallmatrix}})}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Invariants">Invariants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figure-eight_knot_(mathematics)&action=edit&section=4" title="Edit section: Invariants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a> of the figure-eight knot 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 \Delta (t)=-t+3-t^{-1},\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>t</mi> <mo>+</mo> <mn>3</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (t)=-t+3-t^{-1},\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9deecbb76191e9436400b1dd85eb226bde713663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.961ex; height:3.176ex;" alt="{\displaystyle \Delta (t)=-t+3-t^{-1},\ }"></span></dd></dl> <p>the <a href="/wiki/Alexander_polynomial#Alexander–Conway_polynomial" title="Alexander polynomial">Conway polynomial</a> 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 \nabla (z)=1-z^{2},\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla (z)=1-z^{2},\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9f3ffcf767088beaa9d5b35c1e190a26bbba8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.694ex; height:3.176ex;" alt="{\displaystyle \nabla (z)=1-z^{2},\ }"></span><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></dd></dl> <p>and the <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a> 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 V(q)=q^{2}-q+1-q^{-1}+q^{-2}.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(q)=q^{2}-q+1-q^{-1}+q^{-2}.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b17ce58036c174ddaf1dbe18c86ff2d0c9c9fbf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.931ex; height:3.176ex;" alt="{\displaystyle V(q)=q^{2}-q+1-q^{-1}+q^{-2}.\ }"></span></dd></dl> <p>The symmetry between <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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></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 q^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0231d68e66d89fb25248a5270f83fd5cbd7f8f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.412ex; height:3.009ex;" alt="{\displaystyle q^{-1}}"></span> in the Jones polynomial reflects the fact that the figure-eight knot is achiral. </p> <figure typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Figure-eight_knot.webm/400px--Figure-eight_knot.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="400" height="225" data-durationhint="20" data-mwtitle="Figure-eight_knot.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Figure-eight_knot.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/66/Figure-eight_knot.webm/Figure-eight_knot.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/66/Figure-eight_knot.webm/Figure-eight_knot.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/66/Figure-eight_knot.webm/Figure-eight_knot.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/6/66/Figure-eight_knot.webm" type="video/webm; codecs="vp8, vorbis"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/66/Figure-eight_knot.webm/Figure-eight_knot.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/66/Figure-eight_knot.webm/Figure-eight_knot.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/66/Figure-eight_knot.webm/Figure-eight_knot.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/66/Figure-eight_knot.webm/Figure-eight_knot.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /></video></span><figcaption>Figure-eight knot</figcaption></figure> <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=Figure-eight_knot_(mathematics)&action=edit&section=5" 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-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">A braid is called homogeneous if every generator <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 _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab3208a7d0c634ef720e03ff5a9949e8310edc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}"></span> either occurs always with positive or always with negative sign.</span> </li> </ol></div></div> <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=Figure-eight_knot_(mathematics)&action=edit&section=6" 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"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Listing_knot">"Listing knot - Encyclopedia of Mathematics"</a>. <i>encyclopediaofmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-06-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=encyclopediaofmath.org&rft.atitle=Listing+knot+-+Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FListing_knot&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigure-eight+knot+%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="CITEREFGruber" class="citation web cs1">Gruber, Hermann. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060209202316/http://home.in.tum.de/~gruberh/">"Rational Knots with 4 crossings"</a>. <i>Rational Knots database</i>. Archived from <a rel="nofollow" class="external text" href="https://home.in.tum.de/~gruberh/">the original</a> on 2006-02-09<span class="reference-accessdate">. Retrieved <span class="nowrap">5 May</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Rational+Knots+database&rft.atitle=Rational+Knots+with+4+crossings&rft.aulast=Gruber&rft.aufirst=Hermann&rft_id=http%3A%2F%2Fhome.in.tum.de%2F~gruberh%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigure-eight+knot+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_Thurston2002" class="citation cs2"><a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a> (March 2002), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200727020107/http://library.msri.org/books/gt3m/">"7. Computation of volume"</a>, <a rel="nofollow" class="external text" href="http://library.msri.org/books/gt3m/"><i>The Geometry and Topology of Three-Manifolds</i></a>, p. 165, archived from <a rel="nofollow" class="external text" href="http://library.msri.org/books/gt3m/PDF/7.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2020-07-27<span class="reference-accessdate">, retrieved <span class="nowrap">2020-10-19</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=7.+Computation+of+volume&rft.btitle=The+Geometry+and+Topology+of+Three-Manifolds&rft.pages=165&rft.date=2002-03&rft.au=William+Thurston&rft_id=http%3A%2F%2Flibrary.msri.org%2Fbooks%2Fgt3m%2FPDF%2F7.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigure-eight+knot+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://katlas.org/wiki/4_1">4_1</a>", <i><a href="/wiki/The_Knot_Atlas" title="The Knot Atlas">The Knot Atlas</a></i>.</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=Figure-eight_knot_(mathematics)&action=edit&section=7" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ian_Agol" title="Ian Agol">Ian Agol</a>, <i>Bounds on exceptional Dehn filling</i>, <a href="/wiki/Geometry_%26_Topology" title="Geometry & Topology">Geometry & Topology</a> 4 (2000), 431–449. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1799796">1799796</a></li> <li>Chun Cao and Robert Meyerhoff, <i>The orientable cusped hyperbolic 3-manifolds of minimum volume</i>, Inventiones Mathematicae, 146 (2001), no. 3, 451–478. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1869847">1869847</a></li> <li><a href="/wiki/Marc_Lackenby" title="Marc Lackenby">Marc Lackenby</a>, <i>Word hyperbolic Dehn surgery</i>, <a href="/wiki/Inventiones_Mathematicae" title="Inventiones Mathematicae">Inventiones Mathematicae</a> 140 (2000), no. 2, 243–282. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1756996">1756996</a></li> <li><a href="/wiki/Marc_Lackenby" title="Marc Lackenby">Marc Lackenby</a> and Robert Meyerhoff, <a rel="nofollow" class="external text" href="http://arxiv.org/abs/0808.1176"><i>The maximal number of exceptional Dehn surgeries</i></a>, arXiv:0808.1176</li> <li><a href="/wiki/Robion_Kirby" title="Robion Kirby">Robion Kirby</a>, <a rel="nofollow" class="external text" href="http://math.berkeley.edu/~kirby/problems.ps.gz"><i>Problems in low-dimensional topology</i></a>, (see problem 1.77, due to <a href="/wiki/Cameron_Gordon_(mathematician)" title="Cameron Gordon (mathematician)">Cameron Gordon</a>, for exceptional slopes)</li> <li>William Thurston, <a rel="nofollow" class="external text" href="http://msri.org/publications/books/gt3m/"><i>The Geometry and Topology of Three-Manifolds</i></a>, Princeton University lecture notes (1978–1981).</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=Figure-eight_knot_(mathematics)&action=edit&section=8" 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/4_1">4_1</a>", <i><a href="/wiki/The_Knot_Atlas" title="The Knot Atlas">The Knot Atlas</a></i>. Accessed: 7 May 2013.</li> <li><span class="citation mathworld" id="Reference-Mathworld-Figure_Eight_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/FigureEightKnot.html">"Figure Eight 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=Figure+Eight+Knot&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FFigureEightKnot.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigure-eight+knot+%28mathematics%29" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline 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href="/wiki/Template:Knot_theory" title="Template:Knot theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Knot_theory" title="Template talk:Knot theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Knot_theory" title="Special:EditPage/Template:Knot theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Knot_theory_(knots_and_links)" style="font-size:114%;margin:0 4em"><a href="/wiki/Knot_theory" title="Knot theory">Knot theory</a> (<a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knots</a> and <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">links</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hyperbolic_link" title="Hyperbolic link">Hyperbolic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Figure-eight</a> (4<sub>1</sub>)</li> <li><a href="/wiki/Three-twist_knot" title="Three-twist knot">Three-twist</a> (5<sub>2</sub>)</li> <li><a href="/wiki/Stevedore_knot_(mathematics)" title="Stevedore knot (mathematics)">Stevedore</a> (6<sub>1</sub>)</li> <li><a href="/wiki/6%E2%82%82_knot" class="mw-redirect" title="6₂ knot">6<sub>2</sub></a></li> <li><a href="/wiki/6%E2%82%83_knot" class="mw-redirect" title="6₃ knot">6<sub>3</sub></a></li> <li><a href="/wiki/7%E2%82%84_knot" class="mw-redirect" title="7₄ knot">Endless</a> (7<sub>4</sub>)</li> <li><a href="/wiki/Carrick_mat" title="Carrick mat">Carrick mat</a> (8<sub>18</sub>)</li> <li><a href="/wiki/Perko_pair" title="Perko pair">Perko pair</a> (10<sub>161</sub>)</li> <li><a href="/wiki/Conway_knot" title="Conway knot">Conway knot</a> (11n34)</li> <li><a href="/wiki/Kinoshita%E2%80%93Terasaka_knot" title="Kinoshita–Terasaka knot">Kinoshita–Terasaka knot</a> (11n42)</li> <li><a href="/wiki/(%E2%88%922,3,7)_pretzel_knot" title="(−2,3,7) pretzel knot">(−2,3,7) pretzel</a> (12n242)</li> <li><a href="/wiki/Whitehead_link" title="Whitehead link">Whitehead</a> (5<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a> (6<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">3</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span>)</li> <li><a href="/wiki/L10a140_link" title="L10a140 link">L10a140</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Satellite_knot" title="Satellite knot">Satellite</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composite_knot" class="mw-redirect" title="Composite knot">Composite knots</a> <ul><li><a href="/wiki/Granny_knot_(mathematics)" title="Granny knot (mathematics)">Granny</a></li> <li><a href="/wiki/Square_knot_(mathematics)" title="Square knot (mathematics)">Square</a></li></ul></li> <li><a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Knot sum</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Torus_knot" title="Torus knot">Torus</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Unknot" title="Unknot">Unknot</a> (0<sub>1</sub>)</li> <li><a href="/wiki/Trefoil_knot" title="Trefoil knot">Trefoil</a> (3<sub>1</sub>)</li> <li><a href="/wiki/Cinquefoil_knot" title="Cinquefoil knot">Cinquefoil</a> (5<sub>1</sub>)</li> <li><a href="/wiki/7%E2%82%81_knot" class="mw-redirect" title="7₁ knot">Septafoil</a> (7<sub>1</sub>)</li> <li><a href="/wiki/Unlink" title="Unlink">Unlink</a> (0<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Hopf_link" title="Hopf link">Hopf</a> (2<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Solomon%27s_knot" title="Solomon's knot">Solomon's</a> (4<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Knot_invariant" title="Knot invariant">Invariants</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternating_knot" title="Alternating knot">Alternating</a></li> <li><a href="/wiki/Arf_invariant_of_a_knot" title="Arf invariant of a knot">Arf invariant</a></li> <li><a href="/wiki/Bridge_number" title="Bridge number">Bridge no.</a> <ul><li><a href="/wiki/2-bridge_knot" title="2-bridge knot">2-bridge</a></li></ul></li> <li><a href="/wiki/Brunnian_link" title="Brunnian link">Brunnian</a></li> <li><a href="/wiki/Chiral_knot" title="Chiral knot">Chirality</a> <ul><li><a href="/wiki/Invertible_knot" title="Invertible knot">Invertible</a></li></ul></li> <li><a href="/wiki/Crosscap_number" title="Crosscap number">Crosscap no.</a></li> <li><a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">Crossing no.</a></li> <li><a href="/wiki/Finite_type_invariant" title="Finite type invariant">Finite type invariant</a></li> <li><a href="/wiki/Hyperbolic_volume" title="Hyperbolic volume">Hyperbolic volume</a></li> <li><a href="/wiki/Khovanov_homology" title="Khovanov homology">Khovanov homology</a></li> <li><a href="/wiki/Knot_genus" class="mw-redirect" title="Knot genus">Genus</a></li> <li><a href="/wiki/Knot_group" title="Knot group">Knot group</a></li> <li><a href="/wiki/Link_group" title="Link group">Link group</a></li> <li><a href="/wiki/Linking_number" title="Linking number">Linking no.</a></li> <li><a href="/wiki/Knot_polynomial" title="Knot polynomial">Polynomial</a> <ul><li><a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander</a></li> <li><a href="/wiki/Bracket_polynomial" title="Bracket polynomial">Bracket</a></li> <li><a href="/wiki/HOMFLY_polynomial" title="HOMFLY polynomial">HOMFLY</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones</a></li> <li><a href="/wiki/Kauffman_polynomial" title="Kauffman polynomial">Kauffman</a></li></ul></li> <li><a href="/wiki/Pretzel_link" title="Pretzel link">Pretzel</a></li> <li><a href="/wiki/Prime_knot" title="Prime knot">Prime</a> <ul><li><a href="/wiki/List_of_prime_knots" title="List of prime knots">list</a></li></ul></li> <li><a href="/wiki/Stick_number" title="Stick number">Stick no.</a></li> <li><a href="/wiki/Tricolorability" title="Tricolorability">Tricolorability</a></li> <li><a href="/wiki/Unknotting_number" title="Unknotting number">Unknotting no.</a> and <a href="/wiki/Unknotting_problem" title="Unknotting problem">problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation<br />and <a href="/wiki/Knot_operation" title="Knot operation">operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander%E2%80%93Briggs_notation" class="mw-redirect" title="Alexander–Briggs notation">Alexander–Briggs notation</a></li> <li><a href="/wiki/Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation</a></li> <li><a href="/wiki/Dowker%E2%80%93Thistlethwaite_notation" title="Dowker–Thistlethwaite notation">Dowker–Thistlethwaite notation</a></li> <li><a href="/wiki/Flype" title="Flype">Flype</a></li> <li><a href="/wiki/Mutation_(knot_theory)" title="Mutation (knot theory)">Mutation</a></li> <li><a href="/wiki/Reidemeister_move" title="Reidemeister move">Reidemeister move</a></li> <li><a href="/wiki/Skein_relation" title="Skein relation">Skein relation</a></li> <li><a href="/wiki/Knot_tabulation" title="Knot tabulation">Tabulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander%27s_theorem" title="Alexander's theorem">Alexander's theorem</a></li> <li><a href="/wiki/Berge_knot" title="Berge knot">Berge</a></li> <li><a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">Braid theory</a></li> <li><a href="/wiki/Conway_sphere" title="Conway sphere">Conway sphere</a></li> <li><a href="/wiki/Knot_complement" title="Knot complement">Complement</a></li> <li><a href="/wiki/Double_torus_knot" class="mw-redirect" title="Double torus knot">Double torus</a></li> <li><a href="/wiki/Fibered_knot" title="Fibered knot">Fibered</a></li> <li><a href="/wiki/Knot" title="Knot">Knot</a></li> <li><a href="/wiki/List_of_mathematical_knots_and_links" title="List of mathematical knots and links">List of knots and links</a></li> <li><a href="/wiki/Ribbon_knot" title="Ribbon knot">Ribbon</a></li> <li><a href="/wiki/Slice_knot" title="Slice knot">Slice</a></li> <li><a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Sum</a></li> <li><a href="/wiki/Tait_conjectures" title="Tait conjectures">Tait conjectures</a></li> <li><a href="/wiki/Twist_knot" title="Twist knot">Twist</a></li> <li><a href="/wiki/Wild_knot" title="Wild knot">Wild</a></li> <li><a href="/wiki/Writhe" title="Writhe">Writhe</a></li> <li><a href="/wiki/Surgery_theory" title="Surgery theory">Surgery theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> 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