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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Rules_of_tricolorability" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rules_of_tricolorability"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Rules of tricolorability</span> </div> </a> <ul id="toc-Rules_of_tricolorability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div 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class="vector-toc-numb">2.2</span> <span>Example of a non-tricolorable knot</span> </div> </a> <ul id="toc-Example_of_a_non-tricolorable_knot-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Isotopy_invariant" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isotopy_invariant"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Isotopy invariant</span> </div> </a> <ul id="toc-Isotopy_invariant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-In_torus_knots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_torus_knots"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>In torus knots</span> </div> </a> <ul id="toc-In_torus_knots-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sources</span> </div> </a> <ul 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to 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">Property in knot theory</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Tricoloring.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Tricoloring.png/220px-Tricoloring.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Tricoloring.png/330px-Tricoloring.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Tricoloring.png/440px-Tricoloring.png 2x" data-file-width="1800" data-file-height="1800" /></a><figcaption>A tricolored <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knot</a>.</figcaption></figure><p>In the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> field of <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>, the <b>tricolorability</b> of a <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot</a> is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an <a href="/wiki/Knot_invariant" title="Knot invariant">isotopy invariant</a>, and hence can be used to distinguish between two different (non-<a href="/wiki/Regular_isotopy" title="Regular isotopy">isotopic</a>) knots. In particular, since the <a href="/wiki/Unknot" title="Unknot">unknot</a> is not tricolorable, any tricolorable knot is necessarily nontrivial. </p><meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Rules_of_tricolorability">Rules of tricolorability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=1" title="Edit section: Rules of tricolorability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In these rules a <b>strand</b> in a <a href="/wiki/Knot_diagram" class="mw-redirect" title="Knot diagram">knot diagram</a> will be a piece of the string that goes from one undercrossing to the next.<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> A knot is <b>tricolorable</b> if each strand of the knot diagram can be colored one of three colors, subject to the following rules:<sup id="cite_ref-Weisstein_2-0" class="reference"><a href="#cite_note-Weisstein-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>1. At least two colors must be used, and</dd> <dd>2. At each crossing, the three incident strands are either all the same color or all different colors.</dd></dl> <p>Some references state instead that all three colors must be used.<sup id="cite_ref-GP_3-0" class="reference"><a href="#cite_note-GP-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> For a knot, this is equivalent to the definition above; however, for a link it is not. </p><p>"The trefoil knot and trivial 2-link are tricolorable, but the unknot, <a href="/wiki/Whitehead_link" title="Whitehead link">Whitehead link</a>, and <a href="/wiki/Figure-eight_knot_(mathematics)" title="Figure-eight knot (mathematics)">figure-eight knot</a> are not. If the projection of a knot is tricolorable, then <a href="/wiki/Reidemeister_moves" class="mw-redirect" title="Reidemeister moves">Reidemeister moves</a> on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is."<sup id="cite_ref-Weisstein_2-1" class="reference"><a href="#cite_note-Weisstein-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here is an example of how to <a href="/wiki/Color" title="Color">color</a> a knot in accordance of the rules of tricolorability. By convention, knot theorists use the colors red, green, and blue. </p> <div class="mw-heading mw-heading3"><h3 id="Example_of_a_tricolorable_knot">Example of a tricolorable knot</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=3" title="Edit section: Example of a tricolorable knot"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:TricoloredGrannyKnot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/TricoloredGrannyKnot.svg/138px-TricoloredGrannyKnot.svg.png" decoding="async" width="138" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/TricoloredGrannyKnot.svg/207px-TricoloredGrannyKnot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/TricoloredGrannyKnot.svg/276px-TricoloredGrannyKnot.svg.png 2x" data-file-width="138" data-file-height="132" /></a></span> </p><p>The <a href="/wiki/Granny_knot_(mathematics)" title="Granny knot (mathematics)">granny knot</a> is tricolorable. In this coloring the three strands at every crossing have three different colors. Coloring one but not both of the <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil knots</a> all red would also give an admissible coloring. The true lover's knot is also tricolorable.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Tricolorable knots with less than nine crossings include 6<sub>1</sub>, 7<sub>4</sub>, 7<sub>7</sub>, 8<sub>5</sub>, 8<sub>10</sub>, 8<sub>11</sub>, 8<sub>15</sub>, 8<sub>18</sub>, 8<sub>19</sub>, 8<sub>20</sub>, and 8<sub>21</sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Example_of_a_non-tricolorable_knot">Example of a non-tricolorable knot</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=4" title="Edit section: Example of a non-tricolorable knot"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Failed_Tricoloring_of_Figure_8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Failed_Tricoloring_of_Figure_8.svg/140px-Failed_Tricoloring_of_Figure_8.svg.png" decoding="async" width="140" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Failed_Tricoloring_of_Figure_8.svg/210px-Failed_Tricoloring_of_Figure_8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Failed_Tricoloring_of_Figure_8.svg/280px-Failed_Tricoloring_of_Figure_8.svg.png 2x" data-file-width="140" data-file-height="150" /></a></span> </p><p>The <a href="/wiki/Figure-eight_knot_(mathematics)" title="Figure-eight knot (mathematics)">figure-eight knot</a> is not tricolorable. In the diagram shown, it has four strands with each pair of strands meeting at some crossing. If three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of these four strands must have a distinct color. Since tricolorability is a knot invariant, none of its other diagrams can be tricolored either. </p> <div class="mw-heading mw-heading2"><h2 id="Isotopy_invariant">Isotopy invariant</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=5" title="Edit section: Isotopy invariant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tricolorability is an <a href="/wiki/Knot_invariant" title="Knot invariant">isotopy invariant</a>, which is a property of a knot or <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">link</a> that remains constant regardless of any <a href="/wiki/Ambient_isotopy" title="Ambient isotopy">ambient isotopy</a>. This can be proven for tame knots by examining <a href="/wiki/Reidemeister_move" title="Reidemeister move">Reidemeister moves</a>. Since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant of tame knots.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <table border="1" style="border-collapse: collapse;" cellspacing="10" cellpadding="5" align="center"> <tbody><tr> <th>Reidemeister Move I is tricolorable.</th> <th>Reidemeister Move II is tricolorable.</th> <th>Reidemeister Move III is tricolorable. </th></tr> <tr> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Tricolor_Invariance_on_Reidemeister_I.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Tricolor_Invariance_on_Reidemeister_I.svg/126px-Tricolor_Invariance_on_Reidemeister_I.svg.png" decoding="async" width="126" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Tricolor_Invariance_on_Reidemeister_I.svg/188px-Tricolor_Invariance_on_Reidemeister_I.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Tricolor_Invariance_on_Reidemeister_I.svg/251px-Tricolor_Invariance_on_Reidemeister_I.svg.png 2x" data-file-width="300" data-file-height="239" /></a><figcaption></figcaption></figure></td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Tricolor_Invariance_on_Reidemeister_II.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Tricolor_Invariance_on_Reidemeister_II.svg/137px-Tricolor_Invariance_on_Reidemeister_II.svg.png" decoding="async" width="137" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Tricolor_Invariance_on_Reidemeister_II.svg/206px-Tricolor_Invariance_on_Reidemeister_II.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Tricolor_Invariance_on_Reidemeister_II.svg/274px-Tricolor_Invariance_on_Reidemeister_II.svg.png 2x" data-file-width="430" data-file-height="314" /></a><figcaption></figcaption></figure></td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Tricolor_Invariance_on_Reidemeister_III.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Tricolor_Invariance_on_Reidemeister_III.svg/217px-Tricolor_Invariance_on_Reidemeister_III.svg.png" decoding="async" width="217" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Tricolor_Invariance_on_Reidemeister_III.svg/325px-Tricolor_Invariance_on_Reidemeister_III.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Tricolor_Invariance_on_Reidemeister_III.svg/433px-Tricolor_Invariance_on_Reidemeister_III.svg.png 2x" data-file-width="950" data-file-height="439" /></a><figcaption></figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=6" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because tricolorability is a binary classification (a link is either tricolorable or not*), it is a relatively weak invariant. The composition of a tricolorable knot with another knot is always tricolorable. A way to strengthen the invariant is to count the number of possible 3-colorings. In this case, the rule that at least two colors are used is relaxed and now every link has at least three 3-colorings (just color every arc the same color). In this case, a link is 3-colorable if it has more than three 3-colorings. </p><p>Any separable link with a tricolorable separable component is also tricolorable. </p> <div class="mw-heading mw-heading3"><h3 id="In_torus_knots">In torus knots</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=7" title="Edit section: In torus knots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the <a href="/wiki/Torus_knot" title="Torus knot">torus knot</a>/link denoted by (m,n) is tricolorable, then so are (j*m,i*n) and (i*n,j*m) for any natural numbers i and j. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fox_n-coloring" title="Fox n-coloring">Fox n-coloring</a></li> <li><a href="/wiki/Graph_coloring" title="Graph coloring">Graph coloring</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=9" title="Edit section: Sources"><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">Xaoyu Qiao, E. L., <a rel="nofollow" class="external text" href="http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_problem_solving_w2015/Tricolorability.pdf">Knot Theory Week 2: Tricolorability</a> (January 20, 2015), Section 3.</span> </li> <li id="cite_note-Weisstein-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Weisstein_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Weisstein_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Weisstein, Eric W. (2010). <i>CRC Concise Encyclopedia of Mathematics</i>, Second Edition, p.3045. <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781420035223" title="Special:BookSources/9781420035223">9781420035223</a>. quoted at <span class="citation mathworld" id="Reference-Mathworld-Tricolorable"><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/Tricolorable.html">"Tricolorable"</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=Tricolorable&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTricolorable.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATricolorability" class="Z3988"></span></span> Accessed: May 5, 2013.</span> </li> <li id="cite_note-GP-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-GP_3-0">^</a></b></span> <span class="reference-text">Gilbert, N.D. and Porter, T. (1994) <i>Knots and Surfaces</i>, p. 8</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"><a href="/wiki/Mladen_Bestvina" title="Mladen Bestvina">Bestvina, Mladen</a> (February 2003). "<a rel="nofollow" class="external text" href="http://www.math.utah.edu/~bestvina/HS07/knots-hs07.pdf">Knots: a handout for mathcircles</a>", <i>Math.Utah.edu</i>.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdams2004" class="citation book cs1">Adams, Colin (2004). <a rel="nofollow" class="external text" href="https://www.worldcat.org/title/ocm55633800"><i>The knot book: an elementary introduction to the mathematical theory of knots</i></a>. Providence, R.I: American Mathematical Society. pp. 22–27. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3678-1" title="Special:BookSources/978-0-8218-3678-1"><bdi>978-0-8218-3678-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/55633800">55633800</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+knot+book%3A+an+elementary+introduction+to+the+mathematical+theory+of+knots&rft.place=Providence%2C+R.I&rft.pages=22-27&rft.pub=American+Mathematical+Society&rft.date=2004&rft_id=info%3Aoclcnum%2Focm55633800&rft.isbn=978-0-8218-3678-1&rft.aulast=Adams&rft.aufirst=Colin&rft_id=https%3A%2F%2Fwww.worldcat.org%2Ftitle%2Focm55633800&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATricolorability" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tricolorability&action=edit&section=10" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Three-Colorable_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/Three-ColorableKnot.html">"Three-Colorable 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=Three-Colorable+Knot&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FThree-ColorableKnot.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATricolorability" class="Z3988"></span></span> Accessed: May 5, 2013.</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 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 class="mw-selflink selflink">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 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