CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> colorable knot in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> colorable knot </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <h1 id='colourability'>Colourability</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#colourability_2'><math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourability</a></li> <li><a href='#colourability_3'><math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-colourability</a></li> <li><a href='#coloring_by_a_quandle'>Coloring by a quandle</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id='idea'>Idea</h2> <p>The colourability of a knot tells one information about its <a class='existingWikiWord' href='/nlab/show/knot+group'>knot group</a> yet has a simple, and visually attractive aspect that seems almost to avoid all mention of groups, presentations, etc., except at a fairly naive level.</p> <p>The easiest form of colourability to examine is <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourability.</p> <h2 id='colourability_2'><math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourability</h2> <div class='un_defn'> <h6 id='definition'>Definition</h6> <p>A <a class='existingWikiWord' href='/nlab/show/knot+diagram'>knot diagram</a> is <em><math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourable</em> if we can assign colours to its arcs such that</p> <ol> <li> <p>each arc is assigned one colour;</p> </li> <li> <p><em>exactly</em> three colours are used in the assignment;</p> </li> <li> <p>at each crossing, either all the arcs have the same colour, or arcs of all three colours meet in the crossing.</p> </li> </ol> </div> <div class='un_example'> <h6 id='examples_and_nonexamples'>Examples and non-examples</h6> <ul> <li> <p>The usual diagram for the trefoil knot is <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourable. (Just do it! Each arc is given a separate colour and it works.)</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/figure-8+knot'>figure-8 knot</a> diagram</p> <div style=''><svg height='172.61314pt' viewBox='-79.39394 -93.21921 158.78787 172.61314 ' width='158.78787pt' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink'><g transform='translate(0, 79.39394 ) scale(1,-1) translate(0,93.21921 )'><g><g stroke='rgb(0.0%,0.0%,0.0%)'><g fill='rgb(0.0%,0.0%,0.0%)'><g stroke-width='0.4pt'><g></g><g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><g><g><g transform='matrix(0.7071,0.7071,-0.7071,0.7071,-28.45274,28.45274)'><g fill='rgb(100.0%,0.0%,0.0%)'><text font-size='10' style='stroke: none;' text-anchor='middle' transform='scale(1,-1) translate(0.0,0)'></text></g></g></g></g></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><g><g><g transform='matrix(0.7071,-0.7071,0.7071,0.7071,28.45274,28.45274)'><g fill='rgb(100.0%,0.0%,0.0%)'><text font-size='10' style='stroke: none;' text-anchor='middle' transform='scale(1,-1) translate(0.0,0)'></text></g></g></g></g></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><g><g><g transform='matrix(1.0,0.0,0.0,1.0,0.0,-28.45274)'><g fill='rgb(100.0%,0.0%,0.0%)'><text font-size='10' style='stroke: none;' text-anchor='middle' transform='scale(1,-1) translate(0.0,0)'></text></g></g></g></g></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><g><g><g transform='matrix(1.0,0.0,0.0,1.0,0.0,-56.90549)'><g fill='rgb(100.0%,0.0%,0.0%)'><text font-size='10' style='stroke: none;' text-anchor='middle' transform='scale(1,-1) translate(0.0,0)'></text></g></g></g></g></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><path d=' M -2.2071 -54.69838 C -8.82843 -48.07706 -8.82843 -37.28117 0.0 -28.45274 ' style='fill: none;'></path></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><path d=' M 0.0 -56.90549 C 8.82843 -48.07706 8.82843 -37.28117 2.2071 -30.65985 ' style='fill: none;'></path></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><path d=' M -2.20717 -26.24564 C -17.65686 -10.79588 -28.45274 19.08879 -28.45274 28.45274 ' style='fill: none;'></path></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><path d=' M 0.0 -28.45274 C 17.65686 -10.79588 28.45274 19.08879 28.45274 25.33144 ' style='fill: none;'></path></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><path d=' M -28.45274 28.45274 C -28.45274 78.39394 28.45274 78.39394 28.4527 31.5741 ' style='fill: none;'></path></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><path d=' M 2.20717 -59.1126 C 35.31372 -92.21921 78.39394 28.45274 28.45274 28.45274 ' style='fill: none;'></path></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><path d=' M 0.0 -56.90549 C -35.31372 -92.21921 -78.39394 28.45274 -31.5741 28.4527 ' style='fill: none;'></path></g></g></g></g><g><g stroke='rgb(100.0%,0.0%,0.0%)'><g fill='rgb(100.0%,0.0%,0.0%)'><g stroke-width='2.0pt'><path d=' M -25.33133 28.45285 L 28.45274 28.45274 ' style='fill: none;'></path></g></g></g></g></g><g></g></g></g></g></g></g></svg> <p><div class='property'>category: <a class='category_link' href='/nlab/all_pages/svg'>svg</a></div></p> </div> <p>is not <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourable. (Try it!)</p> </li> </ul> </div> <div class='un_theorem'> <h6 id='theorem'>Theorem</h6> <p><math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourability is a <span class='newWikiWord'>knot invariant<a href='/nlab/new/knot++invariant'>?</a></span>.</p> </div> <p>The proof is amusing to work out oneself. You have to show that if a knot diagram <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourable and you perform a <a class='existingWikiWord' href='/nlab/show/Reidemeister+move'>Reidemeister move</a> on it then the result is also <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourable. The thing to note is that any arcs that leave the locality of the move must be coloured the same before and after the move is done.</p> <div class='un_note'> <h6 id='notes'>Notes</h6> <ul> <li> <p>We can now use phrases such as ‘the trefoil knot is <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourable’ as its validity does not depend on what diagram is used to represent it, (by the above and by <a class='existingWikiWord' href='/nlab/show/Reidemeister+moves'>Reidemeister's theorem</a>.)</p> </li> <li> <p>As the trefoil knot is <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourable and the unknot is not, <em>non-trivial knots exist</em>. Moreover, the trefoil is <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-colourable and the figure <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>8</mn></mrow><annotation encoding='application/x-tex'>8</annotation></semantics></math> is not, so these are different. We also get that the <a class='existingWikiWord' href='/nlab/show/bridge+number'>bridge number</a> of the trefoil is <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>, as this provides the missing piece of the argument found in that entry.</p> </li> </ul> </div> <p>There are two comments to make here. First what does this all mean at a deeper topological level? The other is : why stop at <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>? What about <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-colourability? We will handle the second one first.</p> <h2 id='colourability_3'><math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-colourability</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> be an integer - in practice <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n = 1</annotation></semantics></math> or 2 are not that interesting, so usually <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>n \geq 3</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℤ</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_n</annotation></semantics></math> be the additive group of integers modulo <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>.</p> <div class='un_defn'> <h6 id='definition_2'>Definition</h6> <p>An <em><math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-colouring</em> of a knot diagram, <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, is an assignment to each arc of an element of <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℤ</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_n</annotation></semantics></math> in such a way that, at each crossing, the sum of the values assigned on the underpass arcs is twice that on the overpass, and such that in the assignment at least two elements of <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℤ</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_n</annotation></semantics></math> are used.</p> </div> <svg height='101.35714' id='svg2' version='1.0' width='89.928581' xmlns='http://www.w3.org/2000/svg'> <defs id='defs4'> <inkscape:perspective id='perspective10' inkscape:persp3d-origin='372.04724 : 350.78739 : 1' inkscape:vp_x='0 : 526.18109 : 1' inkscape:vp_y='0 : 1000 : 0' inkscape:vp_z='744.09448 : 526.18109 : 1' sodipodi:type='inkscape:persp3d'></inkscape:perspective> </defs> <sodipodi:namedview bordercolor='#666666' borderopacity='1.0' gridtolerance='10000' guidetolerance='10' id='base' inkscape:current-layer='layer1' inkscape:cx='47.132334' inkscape:cy='42.148486' inkscape:document-units='px' inkscape:pageopacity='0.0' inkscape:pageshadow='2' inkscape:window-height='701' inkscape:window-width='640' inkscape:window-x='0' inkscape:window-y='22' inkscape:zoom='2.8' objecttolerance='10' pagecolor='#ffffff' showgrid='false'></sodipodi:namedview> <metadata id='metadata7'> <rdf:RDF> <cc:Work rdf:about=''> <dc:format>image/svg+xml</dc:format> <dc:type rdf:resource='http://purl.org/dc/dcmitype/StillImage'></dc:type> </cc:Work> </rdf:RDF> </metadata> <g id='layer1' transform='translate(-116.28571,-149.30877)'> <path d='M 117.14286,149.80877 C 202.85714,258.3802 205.71429,249.80877 205.71429,249.80877' id='path2383' style='fill: none; fill-rule: evenodd; stroke: #000000; stroke-width: 1px; stroke-linecap: butt; stroke-linejoin: miter; stroke-opacity: 1;'></path> <path d='M 116.78571,250.16591 L 156.78571,206.23734' id='path2385' style='fill: none; fill-rule: evenodd; stroke: #000000; stroke-width: 1px; stroke-linecap: butt; stroke-linejoin: miter; stroke-opacity: 1;'></path> <path d='M 163.21429,200.8802 L 203.92857,153.02305' id='path2387' style='fill: none; fill-rule: evenodd; stroke: #000000; stroke-width: 1px; stroke-linecap: butt; stroke-linejoin: miter; stroke-opacity: 1;'></path> <text id='text2389' style='font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; text-align: start; line-height: 125%; fill: #000000; fill-opacity: 1; stroke: none; stroke-width: 1px; stroke-linecap: butt; stroke-linejoin: miter; stroke-opacity: 1; font-family: Times New Roman;' x='129.64285' xml:space='preserve' y='222.66591'><tspan id='tspan2391' x='129.64285' y='222.66591'>a</tspan></text> <text id='text2393' style='font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; text-align: start; line-height: 125%; fill: #000000; fill-opacity: 1; stroke: none; stroke-width: 1px; stroke-linecap: butt; stroke-linejoin: miter; stroke-opacity: 1; font-family: Times New Roman;' x='176.42857' xml:space='preserve' y='175.52306'><tspan id='tspan2395' x='176.42857' y='175.52306'>c</tspan></text> <text id='text2397' style='font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; text-align: start; line-height: 125%; fill: #000000; fill-opacity: 1; stroke: none; stroke-width: 1px; stroke-linecap: butt; stroke-linejoin: miter; stroke-opacity: 1; font-family: Times New Roman;' x='146.42857' xml:space='preserve' y='178.73734'><tspan id='tspan2399' x='146.42857' y='178.73734'>b</tspan></text> </g> </svg> <p>with <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>+</mo><mi>c</mi><msub><mo>≡</mo> <mi>n</mi></msub><mn>2</mn><mi>b</mi></mrow><annotation encoding='application/x-tex'>a+c \equiv_n 2b</annotation></semantics></math>.</p> <p>What, of course, needs to be checked (left to ‘the reader’) is</p> <ul> <li> <p>this notion is a generalisation of 3-colourability (i.e. coincides in the case of <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>n = 3</annotation></semantics></math>);</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-colourability is preserved under <a class='existingWikiWord' href='/nlab/show/Reidemeister+moves'>Reidemeister moves</a> so can be applied to a knot, not just to a knot diagram;</p> </li> </ul> <p>and then to find some examples of, say, 5-colourability. What knots are 5-colourable? Which <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(2,k)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/torus+knots'>torus knots</a> are 5-colorable, and so on. This is not important mathematically, but is quite fun and, in fact, does give insight and experience in working with the Reidemeister moves.</p> <h2 id='coloring_by_a_quandle'>Coloring by a quandle</h2> <p>There is an even more general notion of coloring a knot <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> by the elements of a <a class='existingWikiWord' href='/nlab/show/quandle'>quandle</a> <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Q</mi><mo>,</mo><mo>⊳</mo><mo>:</mo><mi>Q</mi><mo>×</mo><mi>Q</mi><mo>→</mo><mi>Q</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Q,\rhd : Q \times Q \to Q)</annotation></semantics></math>. Formally, a coloring of <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> by <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> corresponds to a quandle homomorphism from the fundamental quandle <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Q(K)</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math>. Concretely, this says that at each crossing with arcs labelled <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> (as in the above diagram), the identity <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>=</mo><mi>a</mi><mo>⊳</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>c = a \rhd b</annotation></semantics></math> must be respected. In particular, <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-coloring corresponds to coloring by the set <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℤ</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_n</annotation></semantics></math> equipped with the quandle operation <math class='maruku-mathml' display='inline' id='mathml_97d871a90076ba9ee1a41044592d45e7c6b1d091_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>⊳</mo><mi>b</mi><mo>=</mo><mn>2</mn><mi>b</mi><mo>−</mo><mi>a</mi><mspace width='thinmathspace'></mspace><mo stretchy='false'>(</mo><mi>mod</mi><mspace width='thinmathspace'></mspace><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>a \rhd b = 2b - a\,(mod\,n)</annotation></semantics></math>, known as the dihedral quandle.</p> <h2 id='references'>References</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Ralph+Fox'>Ralph Fox</a>, <em>A quick trip through Knot theory</em> <a href='(http://homepages.math.uic.edu/~kauffman/QuickTrip.pdf'>pdf file</a></p> </li> <li> <p>Wikipedia articles on <a href='http://en.wikipedia.org/wiki/Tricolorability'>tricolorability</a> and <a href='http://en.wikipedia.org/wiki/Fox_n-coloring'>Fox n-coloring</a></p> </li> </ul> <div class='property'>category: <a class='category_link' href='/nlab/all_pages/knot+theory'>knot theory</a></div> </div> <div class="revisedby"> <p> Last revised on August 23, 2014 at 22:23:42. See the <a href="/nlab/history/colorable+knot" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/colorable+knot" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/colorable+knot/16" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/colorable+knot" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/colorable+knot" accesskey="S" class="navlink" id="history" rel="nofollow">History (16 revisions)</a> <a href="/nlab/show/colorable+knot/cite" style="color: black">Cite</a> <a href="/nlab/print/colorable+knot" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/colorable+knot" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>