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sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <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"></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">Type of mathematical knot</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ribbon_knot_8_20.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Ribbon_knot_8_20.jpg/220px-Ribbon_knot_8_20.jpg" decoding="async" width="220" height="332" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Ribbon_knot_8_20.jpg/330px-Ribbon_knot_8_20.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Ribbon_knot_8_20.jpg/440px-Ribbon_knot_8_20.jpg 2x" data-file-width="654" data-file-height="988" /></a><figcaption>A 3-dimensional rendering of the ribbon knot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8_{20}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8_{20}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5d6a6a670be4b01f0509fb5d9e2be5991d6c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.039ex; height:2.509ex;" alt="{\displaystyle 8_{20}}"></span>, showing the ribbon property</figcaption></figure> <p>In the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> area of <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>, a <b>ribbon knot</b> is a <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot</a> that bounds a self-intersecting disk with only <i>ribbon singularities</i>. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part of the disk through the slit. More precisely, this type of singularity is a closed arc consisting of intersection points of the disk with itself, such that the preimage of this arc consists of two arcs in the disc, one completely in the interior of the disk and the other having its two endpoints on the disk boundary. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Morse-theoretic_formulation">Morse-theoretic formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ribbon_knot&action=edit&section=1" title="Edit section: Morse-theoretic formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A slice disc <i>M</i> is a smoothly embedded <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 D^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1e7d554748b105413b259fdfb8c9587acb9f1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{2}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3787216441d397d9b26987e0587f325c96c105c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{4}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\cap \partial D^{4}=\partial M\subset S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>∩</mo> <mi mathvariant="normal">∂</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>M</mi> <mo>⊂</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\cap \partial D^{4}=\partial M\subset S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d02206089da719a40ac4d5415f51eebce17eb56e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.854ex; height:2.676ex;" alt="{\displaystyle M\cap \partial D^{4}=\partial M\subset S^{3}}"></span>. Consider 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\colon D^{4}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon D^{4}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3426dc644e29850dfd8f2feb11e332c45b5d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:3.009ex;" alt="{\displaystyle f\colon D^{4}\to \mathbb {R} }"></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(x,y,z,w)=x^{2}+y^{2}+z^{2}+w^{2}}"> <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>w</mi> <mo stretchy="false">)</mo> <mo>=</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>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,z,w)=x^{2}+y^{2}+z^{2}+w^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20850a5ece614d3ccffa78a63f8c999ce7fd2400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.508ex; height:3.176ex;" alt="{\displaystyle f(x,y,z,w)=x^{2}+y^{2}+z^{2}+w^{2}}"></span>. By a small isotopy of <i>M</i> one can ensure that <i>f</i> restricts to a <a href="/wiki/Morse_function" class="mw-redirect" title="Morse function">Morse function</a> on <i>M</i>. One says <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 M\subset \partial D^{4}=S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>M</mi> <mo>⊂</mo> <mi mathvariant="normal">∂</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial M\subset \partial D^{4}=S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb99fea31548dd2300a8be157d472de6ebd9c4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.83ex; height:2.676ex;" alt="{\displaystyle \partial M\subset \partial D^{4}=S^{3}}"></span> is a ribbon knot 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 f_{|M}\colon M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>M</mi> </mrow> </msub> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{|M}\colon M\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9469d7ccc1d391448ae4d08c30573ba7cc17b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.324ex; height:3.009ex;" alt="{\displaystyle f_{|M}\colon M\to \mathbb {R} }"></span> has no interior local maxima. </p> <div class="mw-heading mw-heading2"><h2 id="Slice-ribbon_conjecture">Slice-ribbon conjecture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ribbon_knot&action=edit&section=2" title="Edit section: Slice-ribbon conjecture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every ribbon knot is known to be a <a href="/wiki/Slice_knot" title="Slice knot">slice knot</a>. A famous open problem, posed by <a href="/wiki/Ralph_Fox" title="Ralph Fox">Ralph Fox</a> and known as the <b>slice-ribbon conjecture</b>, asks if the converse is true: is every (smoothly) slice knot ribbon? </p><p><a href="#CITEREFLisca2007">Lisca (2007)</a> showed that the conjecture is true for knots of <a href="/wiki/Bridge_number" title="Bridge number">bridge number</a> two. <a href="#CITEREFGreeneJabuka2011">Greene & Jabuka (2011)</a> showed it to be true for three-stranded <a href="/wiki/Pretzel_link" title="Pretzel link">pretzel knots</a> with odd parameters. However, <a href="#CITEREFGompfScharlemannThompson2010">Gompf, Scharlemann & Thompson (2010)</a> suggested that the conjecture might not be true, and provided a family of knots that could be counterexamples to it. The conjecture was further strengthened when a famous potential counter-example, the (2, 1) cable of the figure-eight knot, was shown to be not slice and thereby not a counterexample.<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ribbon_knot&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFox1962" class="citation cs2"><a href="/wiki/Ralph_Fox" title="Ralph Fox">Fox, R. H.</a> (1962), "Some problems in knot theory", <i>Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961)</i>, Englewood Cliffs, New Jersey: Prentice-Hall, pp. 168–176, <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=0140100">0140100</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Some+problems+in+knot+theory&rft.btitle=Topology+of+3-manifolds+and+related+topics+%28Proc.+The+Univ.+of+Georgia+Institute%2C+1961%29&rft.place=Englewood+Cliffs%2C+New+Jersey&rft.pages=168-176&rft.pub=Prentice-Hall&rft.date=1962&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0140100%23id-name%3DMR&rft.aulast=Fox&rft.aufirst=R.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARibbon+knot" class="Z3988"></span>. Reprinted by Dover Books, 2010.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGompfScharlemannThompson2010" class="citation cs2"><a href="/wiki/Robert_Gompf" title="Robert Gompf">Gompf, Robert E.</a>; <a href="/wiki/Martin_Scharlemann" title="Martin Scharlemann">Scharlemann, Martin</a>; <a href="/wiki/Abigail_Thompson" title="Abigail Thompson">Thompson, Abigail</a> (2010), "Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures", <i>Geometry & Topology</i>, <b>14</b> (4): 2305–2347, <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/1103.1601">1103.1601</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/10.2140%2Fgt.2010.14.2305">10.2140/gt.2010.14.2305</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=2740649">2740649</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:58915479">58915479</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geometry+%26+Topology&rft.atitle=Fibered+knots+and+potential+counterexamples+to+the+property+2R+and+slice-ribbon+conjectures&rft.volume=14&rft.issue=4&rft.pages=2305-2347&rft.date=2010&rft_id=info%3Aarxiv%2F1103.1601&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2740649%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A58915479%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2140%2Fgt.2010.14.2305&rft.aulast=Gompf&rft.aufirst=Robert+E.&rft.au=Scharlemann%2C+Martin&rft.au=Thompson%2C+Abigail&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARibbon+knot" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreeneJabuka2011" class="citation cs2">Greene, Joshua; Jabuka, Stanislav (2011), "The slice-ribbon conjecture for 3-stranded pretzel knots", <i>American Journal of Mathematics</i>, <b>133</b> (3): 555–580, <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/0706.3398">0706.3398</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/10.1353%2Fajm.2011.0022">10.1353/ajm.2011.0022</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=2808326">2808326</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:10279100">10279100</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=The+slice-ribbon+conjecture+for+3-stranded+pretzel+knots&rft.volume=133&rft.issue=3&rft.pages=555-580&rft.date=2011&rft_id=info%3Aarxiv%2F0706.3398&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2808326%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10279100%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1353%2Fajm.2011.0022&rft.aulast=Greene&rft.aufirst=Joshua&rft.au=Jabuka%2C+Stanislav&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARibbon+knot" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKauffman1987" class="citation cs2"><a href="/wiki/Louis_Kauffman" title="Louis Kauffman">Kauffman, Louis H.</a> (1987), <i>On Knots</i>, Annals of Mathematics Studies, vol. 115, Princeton, New Jersey: Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08434-3" title="Special:BookSources/0-691-08434-3"><bdi>0-691-08434-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0907872">0907872</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+Knots&rft.place=Princeton%2C+New+Jersey&rft.series=Annals+of+Mathematics+Studies&rft.pub=Princeton+University+Press&rft.date=1987&rft.isbn=0-691-08434-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D907872%23id-name%3DMR&rft.aulast=Kauffman&rft.aufirst=Louis+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARibbon+knot" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLisca2007" class="citation cs2">Lisca, Paolo (2007), "Lens spaces, rational balls and the ribbon conjecture", <i>Geometry & Topology</i>, <b>11</b>: 429–472, <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/0701610">math/0701610</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/10.2140%2Fgt.2007.11.429">10.2140/gt.2007.11.429</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=2302495">2302495</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:15238217">15238217</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geometry+%26+Topology&rft.atitle=Lens+spaces%2C+rational+balls+and+the+ribbon+conjecture&rft.volume=11&rft.pages=429-472&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0701610&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2302495%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15238217%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2140%2Fgt.2007.11.429&rft.aulast=Lisca&rft.aufirst=Paolo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARibbon+knot" class="Z3988"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References_2">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ribbon_knot&action=edit&section=4" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaiKangMallickPark2022" class="citation arxiv cs1">Dai, Irving; Kang, Sungkyung; Mallick, Abhishek; Park, JungHwan; Stoffregen, Matthew (28 July 2022). "The $(2,1)$-cable of the figure-eight knot is not smoothly slice". <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/2207.14187">2207.14187</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.GT">math.GT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+%24%282%2C1%29%24-cable+of+the+figure-eight+knot+is+not+smoothly+slice&rft.date=2022-07-28&rft_id=info%3Aarxiv%2F2207.14187&rft.aulast=Dai&rft.aufirst=Irving&rft.au=Kang%2C+Sungkyung&rft.au=Mallick%2C+Abhishek&rft.au=Park%2C+JungHwan&rft.au=Stoffregen%2C+Matthew&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARibbon+knot" 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="CITEREFSloman2023" class="citation web cs1">Sloman, Leila (2 February 2023). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/mathematicians-prove-this-knot-cannot-solve-major-problem-20230202">"Mathematicians Eliminate Long-Standing Threat to Knot Conjecture"</a>. <i><a href="/wiki/Quanta_Magazine" title="Quanta Magazine">Quanta Magazine</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quanta+Magazine&rft.atitle=Mathematicians+Eliminate+Long-Standing+Threat+to+Knot+Conjecture&rft.date=2023-02-02&rft.aulast=Sloman&rft.aufirst=Leila&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fmathematicians-prove-this-knot-cannot-solve-major-problem-20230202&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARibbon+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=Ribbon_knot&action=edit&section=5" 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="CITEREFSloman2022" class="citation web cs1">Sloman, Leila (18 May 2022). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/how-complex-is-a-knot-new-proof-reveals-ranking-system-that-works-20220518/">"How Complex Is a Knot? New Proof Reveals Ranking System That Works"</a>. <i>Quanta Magazine</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quanta+Magazine&rft.atitle=How+Complex+Is+a+Knot%3F+New+Proof+Reveals+Ranking+System+That+Works&rft.date=2022-05-18&rft.aulast=Sloman&rft.aufirst=Leila&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fhow-complex-is-a-knot-new-proof-reveals-ranking-system-that-works-20220518%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARibbon+knot" class="Z3988"></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 ol,.mw-parser-output .hlist.inline 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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 href="/wiki/Figure-eight_knot_(mathematics)" title="Figure-eight knot (mathematics)">Figure-eight</a> (4<sub>1</sub>)</li> <li><a href="/wiki/Three-twist_knot" title="Three-twist knot">Three-twist</a> (5<sub>2</sub>)</li> <li><a href="/wiki/Stevedore_knot_(mathematics)" title="Stevedore knot (mathematics)">Stevedore</a> (6<sub>1</sub>)</li> <li><a href="/wiki/6%E2%82%82_knot" class="mw-redirect" title="6₂ knot">6<sub>2</sub></a></li> <li><a href="/wiki/6%E2%82%83_knot" class="mw-redirect" title="6₃ knot">6<sub>3</sub></a></li> <li><a href="/wiki/7%E2%82%84_knot" class="mw-redirect" title="7₄ knot">Endless</a> (7<sub>4</sub>)</li> <li><a href="/wiki/Carrick_mat" title="Carrick mat">Carrick mat</a> (8<sub>18</sub>)</li> <li><a href="/wiki/Perko_pair" title="Perko pair">Perko pair</a> (10<sub>161</sub>)</li> <li><a href="/wiki/Conway_knot" title="Conway knot">Conway knot</a> (11n34)</li> <li><a href="/wiki/Kinoshita%E2%80%93Terasaka_knot" title="Kinoshita–Terasaka knot">Kinoshita–Terasaka knot</a> (11n42)</li> <li><a href="/wiki/(%E2%88%922,3,7)_pretzel_knot" title="(−2,3,7) pretzel knot">(−2,3,7) pretzel</a> (12n242)</li> <li><a href="/wiki/Whitehead_link" title="Whitehead link">Whitehead</a> (5<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a> (6<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">3</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span>)</li> <li><a href="/wiki/L10a140_link" title="L10a140 link">L10a140</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Satellite_knot" title="Satellite knot">Satellite</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composite_knot" class="mw-redirect" title="Composite knot">Composite knots</a> <ul><li><a href="/wiki/Granny_knot_(mathematics)" title="Granny knot (mathematics)">Granny</a></li> <li><a href="/wiki/Square_knot_(mathematics)" title="Square knot (mathematics)">Square</a></li></ul></li> <li><a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Knot sum</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Torus_knot" title="Torus knot">Torus</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Unknot" title="Unknot">Unknot</a> (0<sub>1</sub>)</li> <li><a href="/wiki/Trefoil_knot" title="Trefoil knot">Trefoil</a> (3<sub>1</sub>)</li> <li><a href="/wiki/Cinquefoil_knot" title="Cinquefoil knot">Cinquefoil</a> (5<sub>1</sub>)</li> <li><a href="/wiki/7%E2%82%81_knot" class="mw-redirect" title="7₁ knot">Septafoil</a> (7<sub>1</sub>)</li> <li><a href="/wiki/Unlink" title="Unlink">Unlink</a> (0<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Hopf_link" title="Hopf link">Hopf</a> (2<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Solomon%27s_knot" title="Solomon's knot">Solomon's</a> (4<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Knot_invariant" title="Knot invariant">Invariants</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternating_knot" title="Alternating knot">Alternating</a></li> <li><a href="/wiki/Arf_invariant_of_a_knot" title="Arf invariant of a knot">Arf invariant</a></li> <li><a href="/wiki/Bridge_number" title="Bridge number">Bridge no.</a> <ul><li><a href="/wiki/2-bridge_knot" title="2-bridge knot">2-bridge</a></li></ul></li> <li><a href="/wiki/Brunnian_link" title="Brunnian link">Brunnian</a></li> <li><a href="/wiki/Chiral_knot" title="Chiral knot">Chirality</a> <ul><li><a href="/wiki/Invertible_knot" title="Invertible knot">Invertible</a></li></ul></li> <li><a href="/wiki/Crosscap_number" title="Crosscap number">Crosscap no.</a></li> <li><a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">Crossing no.</a></li> <li><a href="/wiki/Finite_type_invariant" title="Finite type invariant">Finite type invariant</a></li> <li><a href="/wiki/Hyperbolic_volume" title="Hyperbolic volume">Hyperbolic volume</a></li> <li><a href="/wiki/Khovanov_homology" title="Khovanov homology">Khovanov homology</a></li> <li><a href="/wiki/Knot_genus" class="mw-redirect" title="Knot genus">Genus</a></li> <li><a href="/wiki/Knot_group" title="Knot group">Knot group</a></li> <li><a href="/wiki/Link_group" title="Link group">Link group</a></li> <li><a href="/wiki/Linking_number" title="Linking number">Linking no.</a></li> <li><a href="/wiki/Knot_polynomial" title="Knot polynomial">Polynomial</a> <ul><li><a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander</a></li> <li><a href="/wiki/Bracket_polynomial" title="Bracket polynomial">Bracket</a></li> <li><a href="/wiki/HOMFLY_polynomial" title="HOMFLY polynomial">HOMFLY</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones</a></li> <li><a href="/wiki/Kauffman_polynomial" title="Kauffman polynomial">Kauffman</a></li></ul></li> <li><a href="/wiki/Pretzel_link" title="Pretzel link">Pretzel</a></li> <li><a href="/wiki/Prime_knot" title="Prime knot">Prime</a> <ul><li><a href="/wiki/List_of_prime_knots" title="List of prime knots">list</a></li></ul></li> <li><a href="/wiki/Stick_number" title="Stick number">Stick no.</a></li> <li><a href="/wiki/Tricolorability" title="Tricolorability">Tricolorability</a></li> <li><a href="/wiki/Unknotting_number" title="Unknotting number">Unknotting no.</a> and <a href="/wiki/Unknotting_problem" title="Unknotting problem">problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation<br />and <a href="/wiki/Knot_operation" title="Knot operation">operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander%E2%80%93Briggs_notation" class="mw-redirect" title="Alexander–Briggs notation">Alexander–Briggs notation</a></li> <li><a href="/wiki/Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation</a></li> <li><a href="/wiki/Dowker%E2%80%93Thistlethwaite_notation" title="Dowker–Thistlethwaite notation">Dowker–Thistlethwaite notation</a></li> <li><a href="/wiki/Flype" title="Flype">Flype</a></li> <li><a href="/wiki/Mutation_(knot_theory)" title="Mutation (knot theory)">Mutation</a></li> <li><a href="/wiki/Reidemeister_move" title="Reidemeister move">Reidemeister move</a></li> <li><a href="/wiki/Skein_relation" title="Skein relation">Skein relation</a></li> <li><a href="/wiki/Knot_tabulation" title="Knot tabulation">Tabulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander%27s_theorem" title="Alexander's theorem">Alexander's theorem</a></li> <li><a href="/wiki/Berge_knot" title="Berge knot">Berge</a></li> <li><a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">Braid theory</a></li> <li><a href="/wiki/Conway_sphere" title="Conway sphere">Conway sphere</a></li> <li><a href="/wiki/Knot_complement" title="Knot complement">Complement</a></li> <li><a href="/wiki/Double_torus_knot" class="mw-redirect" title="Double torus knot">Double torus</a></li> <li><a href="/wiki/Fibered_knot" title="Fibered knot">Fibered</a></li> <li><a href="/wiki/Knot" title="Knot">Knot</a></li> <li><a href="/wiki/List_of_mathematical_knots_and_links" title="List of mathematical knots and links">List of knots and links</a></li> <li><a class="mw-selflink selflink">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 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