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width: 0.3em;"></span> <a href="/nlab/show/diff/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/discussion/18315/#Item_1" title="Discuss this page in its dedicated thread on the nForum" 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"> <p class="show_diff"> Showing changes from revision #8 to #9: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <del class='diffmod'><p>The <strong>trefoil knot</strong> is a famous <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a>. One of the reasons is that in the list of knots, ordered by <a class='existingWikiWord' href='/nlab/show/diff/crossing+number'>crossing number</a>, it is the first ‘real’ knot one meets, being the simplest non-trivial knot. (The first knot listed is usually the ‘unknot’, i.e. the unknotted circle.) The trefoil has crossing number 3.</p></del><ins class='diffmod'><div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='knot_theory'>Knot theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/knot'>knot theory</a></strong></p> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a></strong>, <strong><a class='existingWikiWord' href='/nlab/show/diff/link'>link</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/knot+complement'>knot complement</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/link+diagram'>knot diagrams</a>, <a class='existingWikiWord' href='/nlab/show/diff/chord+diagram'>chord diagram</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Reidemeister+move'>Reidemeister move</a></p> </li> </ul> <p><strong>Examples/classes:</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/trefoil+knot'>trefoil knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/torus+knot'>torus knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/singular+knot'>singular knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+link'>hyperbolic knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Borromean+link'>Borromean link</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+link'>Whitehead link</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hopf+link'>Hopf link</a></p> </li> </ul> <p><strong>Types</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/prime+knot'>prime knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mutant+knot'>mutant knot</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/knot+invariant'>knot invariants</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crossing+number'>crossing number</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bridge+number'>bridge number</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unknotting+number'>unknotting number</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colorable+knot'>colorability</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/knot+group'>knot group</a></p> </li> <li> <p><span class='newWikiWord'>knot genus<a href='/nlab/new/knot+genus'>?</a></span></p> </li> <li> <p>polynomial knot invariants</p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/quantum+observable'>observables</a> of <a class='existingWikiWord' href='/nlab/show/diff/non-perturbative+quantum+field+theory'>non-perturbative</a> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a>)</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jones+polynomial'>Jones polynomial</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/HOMFLY-PT+polynomial'>HOMFLY polynomial</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Alexander+polynomial'>Alexander polynomial</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Reshetikhin-Turaev+construction'>Reshetikhin-Turaev invariants</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Vassiliev+invariant'>Vassiliev knot invariants</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/quantum+observable'>observables</a> of <a class='existingWikiWord' href='/nlab/show/diff/perturbative+quantum+field+theory'>pertrubative</a> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Khovanov+homology'>Khovanov homology</a></p> </li> <li> <p><span class='newWikiWord'>Kauffman bracket<a href='/nlab/new/Kauffman+bracket'>?</a></span></p> </li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/link+invariant'>link invariants</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Milnor+mu-bar+invariant'>Milnor mu-bar invariants</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/linking+number'>linking number</a></p> </li> </ul> <p><strong>Related concepts:</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Vassiliev+skein+relation'>Vassiliev skein relation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Seifert+surface'>Seifert surface</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/virtual+knot+theory'>virtual knot theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dehn+surgery'>Dehn surgery</a>, <a class='existingWikiWord' href='/nlab/show/diff/Kirby+calculus'>Kirby calculus</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/volume+conjecture'>volume conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/arithmetic+topology'>arithmetic topology</a></p> </li> </ul> </div> </div> </div></ins> <p><span><del class='diffmod'> Here</del><ins class='diffmod'> \tableofcontents</ins><del class='diffdel'> is</del><del class='diffdel'> a</del><del class='diffdel'> traditional</del><del class='diffdel'> view:</del></span></p><span /><ins class='diffins'><h2 id='idea'>Idea</h2></ins><ins class='diffins'> </ins><ins class='diffins'><p>The <strong>trefoil knot</strong> is a famous example of a <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a>. In the list of knots, ordered by <a class='existingWikiWord' href='/nlab/show/diff/crossing+number'>crossing number</a>, it is the first ‘real’ knot one meets, being the simplest non-trivial knot. (The first knot listed is usually the ‘<a class='existingWikiWord' href='/nlab/show/diff/unknot'>unknot</a>’, i.e. the unknotted circle.) The trefoil has crossing number 3.</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>Here is the traditional <a class='existingWikiWord' href='/nlab/show/diff/link+diagram'>knot diagram</a> for the trefoil knot:</p></ins><ins class='diffins'> </ins><svg height='128.23235pt' viewBox='-73.88037 -64.76646 147.76074 128.23235 ' width='147.76074pt' xmlns:xlink='http://www.w3.org/1999/xlink' xmlns='http://www.w3.org/2000/svg'> <g transform='translate(0, 63.46588 ) scale(1,-1) translate(0,64.76646 )'> <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> <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> <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(-0.5,0.86603,-0.86603,-0.5,24.64085,14.22636)'> <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> <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(-0.5,-0.86603,0.86603,-0.5,-24.64085,14.22636)'> <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> <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 -26.24564 C -8.82843 -19.62431 -21.40941 2.16649 -24.64085 14.22636 ' style='fill: none;' /> </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 -35.31372 -63.76646 -72.88037 1.30057 -27.65584 13.41844 ' style='fill: none;' /> </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 23.833 11.21141 C 21.40941 2.16649 8.82843 -19.62431 0.0 -28.45274 ' style='fill: none;' /> </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 24.64085 14.22636 C 72.88037 1.30057 35.31372 -63.76646 2.20717 -30.65985 ' style='fill: none;' /> </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 -21.6259 15.03423 C -12.58098 17.45781 12.58098 17.45781 24.64085 14.22636 ' style='fill: none;' /> </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 -24.64085 14.22636 C -37.56665 62.46588 37.56665 62.46588 25.44868 17.24138 ' style='fill: none;' /> </g> </g> </g> </g> </g> <g> </g> </g> </g> </g> </g> </g> </svg> <p><span><del class='diffmod'> Here</del><ins class='diffmod'> \begin{tikzpicture}</ins><del class='diffmod'> is</del><ins class='diffmod'> \foreach</ins><del class='diffmod'> a</del><ins class='diffmod'> \n</ins><del class='diffmod'> depiction</del><ins class='diffmod'> in</ins><del class='diffmod'> with</del><ins class='diffmod'> {0,1,2}</ins><ins class='diffins'> {</ins><ins class='diffins'> \begin{scope}[rotate=\n120-4</ins><ins class='diffins'> ]</ins><ins class='diffins'> \drawline</ins><ins class='diffins'> width=2</ins><ins class='diffins'> (0,-1)</ins><ins class='diffins'> ..</ins><ins class='diffins'> controls</ins><ins class='diffins'> (-1,.2)</ins><ins class='diffins'> and</ins><ins class='diffins'> (-2,2)</ins><ins class='diffins'> ..</ins><ins class='diffins'> (0,2)</ins><ins class='diffins'> ..</ins><ins class='diffins'> controls</ins><ins class='diffins'> (1,2)</ins><ins class='diffins'> and</ins><ins class='diffins'> (1,1)</ins><ins class='diffins'> ..</ins><ins class='diffins'> (.9,.7);</ins><ins class='diffins'> \end{scope}</ins><ins class='diffins'> };</ins><ins class='diffins'> \end{tikzpicture}]</ins></span><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/bridge+number'>bridge number</a></del><del class='diffdel'> 2:</del></p><span /><ins class='diffins'><p>Here is an alternative depiction with <a class='existingWikiWord' href='/nlab/show/diff/bridge+number'>bridge number</a> 2:</p></ins><ins class='diffins'> </ins><svg height='257.06586pt' viewBox='-71.62744 -128.53293 143.25488 257.06586 ' width='143.25488pt' xmlns:xlink='http://www.w3.org/1999/xlink' xmlns='http://www.w3.org/2000/svg'> <g transform='translate(0, 128.53293 ) scale(1,-1) translate(0,128.53293 )'> <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(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'> <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'> <g> <g> <g transform='matrix(1.0,0.0,0.0,1.0,-14.22636,0.0)'> <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(1.0,0.0,0.0,1.0,14.22636,0.0)'> <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 0.0 56.90549 C -70.62744 127.53293 -70.62744 -127.53293 -2.20717 -59.1126 ' style='fill: none;' /> </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 70.62744 127.53293 70.62744 -127.53293 0.0 -56.90549 ' style='fill: none;' /> </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 17.65686 39.24863 31.88322 17.65686 14.22636 0.0 ' style='fill: none;' /> </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 54.69838 C -17.65686 39.24863 -31.88322 17.65686 -16.43353 2.2071 ' style='fill: none;' /> </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 -54.69838 C 17.65686 -39.24863 31.88322 -17.65686 16.43353 -2.2071 ' style='fill: none;' /> </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 -17.65686 -39.24863 -31.88322 -17.65686 -14.22636 0.0 ' style='fill: none;' /> </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 -14.22636 0.0 C -5.39793 8.82843 5.39793 8.82843 12.01926 2.2071 ' style='fill: none;' /> </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 -12.01926 -2.2071 C -5.39793 -8.82843 5.39793 -8.82843 14.22636 0.0 ' style='fill: none;' /> </g> </g> </g> </g> </g> <g> </g> </g> </g> </g> </g> </g> </svg> <ins class='diffins'><p>\begin{remark} To include one of the above <code>svg</code> pictures on a page, write <code>[[!include trefoil knot - SVG]]</code> or <code>[[!include trefoil knot (2 bridge) - SVG]]</code>. \end{remark}</p></ins><ins class='diffins'> </ins><ins class='diffins'><h2 id='related_entries'>Related entries</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unknot'>unknot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/figure+eight+knot'>figure eight knot</a></p> </li> </ul></ins><ins class='diffins'> </ins><ins class='diffins'><h2 id='properties'>Properties</h2></ins><ins class='diffins'> </ins><p>The <a class='existingWikiWord' href='/nlab/show/diff/knot+group'>knot group</a> of the trefoil knot (calculated either by the Dehn or Wirtinger presentations) has two very useful presentations:</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_4bb2115353a57d9d49ad8dc8541a12cfd398e283_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo lspace='mediummathspace' rspace='mediummathspace'>∣</mo><mi>x</mi><mi>y</mi><mi>x</mi><mo>=</mo><mi>y</mi><mi>x</mi><mi>y</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\langle x,y \mid x y x=y x y\rangle</annotation></semantics></math>, which is the <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a>, <strong>Br 3</strong>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_4bb2115353a57d9d49ad8dc8541a12cfd398e283_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>⟨</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>|</mo><msup><mi>a</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>b</mi> <mn>3</mn></msup><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>\langle a,b | a^2= b^3\rangle</annotation></semantics></math>, in which the pair of numbers, <math class='maruku-mathml' display='inline' id='mathml_4bb2115353a57d9d49ad8dc8541a12cfd398e283_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(2,3)</annotation></semantics></math>, is apparent. These reflect the fact that the trefoil is a <math class='maruku-mathml' display='inline' id='mathml_4bb2115353a57d9d49ad8dc8541a12cfd398e283_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(2,3)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/torus+knot'>torus knot</a>. (Of course, it is also a (3,2)-torus knot.)</p> </li> </ul><ins class='diffins'> </ins><ins class='diffins'><p><div class='property'> category: <a class='category_link' href='/nlab/list/knot+theory'>knot theory</a></div> </p></ins> </div> <div class="revisedby"> <p> Last revised on July 18, 2024 at 18:30:24. 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