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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> longitude of a link component </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/discussion/8940/#Item_5" 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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="knot_theory">Knot theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/knot+theory">knot theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/knot">knot</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/link">link</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/isotopy">isotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knot+complement">knot complement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knot+diagrams">knot diagrams</a>, <a class="existingWikiWord" href="/nlab/show/chord+diagram">chord diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reidemeister+move">Reidemeister move</a></p> </li> </ul> <p><strong>Examples/classes:</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trefoil+knot">trefoil knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/torus+knot">torus knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+knot">singular knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperbolic+knot">hyperbolic knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borromean+link">Borromean link</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+link">Whitehead link</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+link">Hopf link</a></p> </li> </ul> <p><strong>Types</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+knot">prime knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mutant+knot">mutant knot</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/knot+invariants">knot invariants</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/crossing+number">crossing number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bridge+number">bridge number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unknotting+number">unknotting number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colorability">colorability</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/quantum+observables">observables</a> of <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jones+polynomial">Jones polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HOMFLY+polynomial">HOMFLY polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+polynomial">Alexander polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+invariants">Reshetikhin-Turaev invariants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vassiliev+knot+invariants">Vassiliev knot invariants</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+observables">observables</a> of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">pertrubative</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/link+invariants">link invariants</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Milnor+mu-bar+invariants">Milnor mu-bar invariants</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linking+number">linking number</a></p> </li> </ul> <p><strong>Related concepts:</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Vassiliev+skein+relation">Vassiliev skein relation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Seifert+surface">Seifert surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+knot+theory">virtual knot theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a>, <a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/volume+conjecture">volume conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/knot+theory">knot theory</a></div></div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#section_Introduction'>Introduction</a></li> <li><a href='#section_Combinatorial_definition'>Combinatorial definition</a></li> </ul> </div> <p><h2 id='section_Introduction'>Introduction</h2></p> <p>The <em>longitude</em> of a <a class="existingWikiWord" href="/nlab/show/knot">knot</a>, or more generally of a component of a <a class="existingWikiWord" href="/nlab/show/link">link</a>, plays a crucial role in the link-theoretic approach to <a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a> by means of the <span class="newWikiWord">Lickorish-Wallace theorem<a href="/nlab/new/Lickorish-Wallace+theorem">?</a></span> and the <a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a>. It can be defined either geometrically or combinatorially.</p> <p><h2 id='section_Combinatorial_definition'>Combinatorial definition</h2></p> <p>The longitude of a knot or of a link component can be defined purely within <a class="existingWikiWord" href="/nlab/show/diagrammatic+knot+theory">diagrammatic knot theory</a>. We describe this in this section.</p> <p> <div class='num_defn' id='DefinitionLongitudeOfALinkComponent'> <h6>Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a component of a <a class="existingWikiWord" href="/nlab/show/link+diagram">link diagram</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>, viewed as defining a link with the <span class="newWikiWord">blackboard framing<a href="/nlab/new/blackboard+framing">?</a></span>. Pick an <span class="newWikiWord">orientation<a href="/nlab/new/orientation+of+a+link+diagram">?</a></span> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>, and pick a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. The <em>longitude</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and the chosen orientation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/word">word</a> in the <a class="existingWikiWord" href="/nlab/show/free+group">free group</a> of the set of <span class="newWikiWord">arcs<a href="/nlab/new/arc+of+a+link+diagram">?</a></span> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> defined inductively as follows.</p> <p>1) Begin at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> with the empty word.</p> <p>2) Walk along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> following the orientation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> until one reaches a crossing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> which one approaches by means of an under-edge (one does not stop at crossings which one approaches and leaves by means of over-edges). If the orientations of the crossings are as follows, add <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> to the end of the word obtained thus far.</p> <div style="text-align: center"> <svg xmlns:xlink="http://www.w3.org/1999/xlink" width="57.291pt" height="68.122pt" viewBox="0 0 57.291 68.122" version="1.1" xmlns="http://www.w3.org/2000/svg"> <defs> <g> <symbol overflow="visible" id="831385639-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="831385639-glyph0-1"> <path style="stroke:none;" d="M 3.71875 -3.765625 C 3.53125 -4.140625 3.25 -4.40625 2.796875 -4.40625 C 1.640625 -4.40625 0.40625 -2.9375 0.40625 -1.484375 C 0.40625 -0.546875 0.953125 0.109375 1.71875 0.109375 C 1.921875 0.109375 2.421875 0.0625 3.015625 -0.640625 C 3.09375 -0.21875 3.453125 0.109375 3.921875 0.109375 C 4.28125 0.109375 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</symbol> </g> </defs> <g id="surface1"> <path style="fill:none;stroke-width:0.59776;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.00046875 28.345219 L -0.00046875 -27.764156 " transform="matrix(1,0,0,-1,28.645,28.646)"></path> <path style="fill:none;stroke-width:0.4782;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.374824 1.831563 C -1.261542 1.144063 0.00017625 0.112813 0.343926 -0.00046875 C 0.00017625 -0.11375 -1.261542 -1.145 -1.374824 -1.8325 " transform="matrix(0,1,1,0,28.645,56.40998)"></path> <path style="fill:none;stroke-width:0.59776;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -28.348125 0.00146875 L -4.980937 0.00146875 " transform="matrix(1,0,0,-1,28.645,28.646)"></path> <path style="fill:none;stroke-width:0.59776;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 4.98 0.00146875 L 27.765156 0.00146875 " transform="matrix(1,0,0,-1,28.645,28.646)"></path> <path style="fill:none;stroke-width:0.4782;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.373824 1.8335 C -1.260542 1.146 0.00117625 0.11475 0.344926 0.00146875 C 0.00117625 -0.115719 -1.260542 -1.146969 -1.373824 -1.834469 " transform="matrix(1,0,0,-1,56.40898,28.646)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#831385639-glyph0-1" x="26.012" y="64.801"></use> </g> </g> </svg> </div> <p>If the orientations of the arcs of the crossing are instead as follows, add <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a^{-1}</annotation></semantics></math> to the end of the word obtained thus far.</p> <div style="text-align: center"> <svg xmlns:xlink="http://www.w3.org/1999/xlink" width="57.291pt" height="68.122pt" viewBox="0 0 57.291 68.122" version="1.1" xmlns="http://www.w3.org/2000/svg"> <defs> <g> <symbol overflow="visible" id="602912419-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="602912419-glyph0-1"> <path style="stroke:none;" d="M 3.71875 -3.765625 C 3.53125 -4.140625 3.25 -4.40625 2.796875 -4.40625 C 1.640625 -4.40625 0.40625 -2.9375 0.40625 -1.484375 C 0.40625 -0.546875 0.953125 0.109375 1.71875 0.109375 C 1.921875 0.109375 2.421875 0.0625 3.015625 -0.640625 C 3.09375 -0.21875 3.453125 0.109375 3.921875 0.109375 C 4.28125 0.109375 4.5 -0.125 4.671875 -0.4375 C 4.828125 -0.796875 4.96875 -1.40625 4.96875 -1.421875 C 4.96875 -1.53125 4.875 -1.53125 4.84375 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style="fill:none;stroke-width:0.59776;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.00046875 27.763187 L -0.00046875 -28.346188 " transform="matrix(1,0,0,-1,28.645,28.646)"></path> <path style="fill:none;stroke-width:0.4782;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.375793 1.8325 C -1.258605 1.145 -0.0007925 0.11375 0.342957 0.00046875 C -0.0007925 -0.112813 -1.258605 -1.144063 -1.375793 -1.831563 " transform="matrix(0,-1,-1,0,28.645,0.88202)"></path> <path style="fill:none;stroke-width:0.59776;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -28.348125 0.00146875 L -4.980937 0.00146875 " transform="matrix(1,0,0,-1,28.645,28.646)"></path> <path style="fill:none;stroke-width:0.59776;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 4.98 0.00146875 L 27.765156 0.00146875 " transform="matrix(1,0,0,-1,28.645,28.646)"></path> <path style="fill:none;stroke-width:0.4782;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.373824 1.8335 C -1.260542 1.146 0.00117625 0.11475 0.344926 0.00146875 C 0.00117625 -0.115719 -1.260542 -1.146969 -1.373824 -1.834469 " transform="matrix(1,0,0,-1,56.40898,28.646)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#602912419-glyph0-1" x="26.012" y="64.801"></use> </g> </g> </svg> </div> <p>The arc <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> is not required to, and may not, belong to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>3) Repeat Step 2) until we return to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>There is an alternative definition if one works with <span class="newWikiWord">framed link diagrams<a href="/nlab/new/framed+link+diagrams">?</a></span>, which involves first replacing one’s original link diagram with one to which a certain number of twists (i.e. <a class="existingWikiWord" href="/nlab/show/R1+move">R1 moves</a>) have been applied according to the framing, and then using Definition <a class="maruku-ref" href="#DefinitionLongitudeOfALinkComponent"></a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>Consider the <a class="existingWikiWord" href="/nlab/show/trefoil">trefoil</a> with a chosen point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and orientation as shown below. 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</g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#948782310-glyph0-2" x="114.083" y="81.305"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#948782310-glyph0-3" x="91.381" y="90.019"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#948782310-glyph0-4" x="61.273" y="57.869"></use> </g> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.19501 1.594263 C -1.095401 0.996607 0.0003025 0.100122 0.299131 0.000513125 C 0.0003025 -0.0990963 -1.095401 -0.995581 -1.19501 -1.593237 " transform="matrix(0,-2,-2,0,127.55962,57.60998)"></path> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.196266 1.593088 C -1.096657 0.995431 -0.00095375 0.0989469 0.299827 -0.0006625 C -0.00095375 -0.100272 -1.096657 -0.996756 -1.196266 -1.594412 " transform="matrix(-2,0,0,2,114.30278,14.1732)"></path> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.196266 1.593516 C -1.096657 0.995859 -0.00095375 0.099375 0.299827 -0.000234375 C -0.00095375 -0.0998437 -1.096657 -0.996328 -1.196266 -1.593984 " transform="matrix(-2,0,0,2,114.30278,42.52)"></path> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.194567 1.593138 C -1.094957 0.995482 0.000745625 0.0989975 0.299574 -0.000611875 C 0.000745625 -0.100221 -1.094957 -0.996706 -1.194567 -1.594362 " transform="matrix(2,0,0,-2,126.64304,99.21362)"></path> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.196176 1.593556 C -1.096566 0.9959 -0.000863125 0.0994156 0.297965 -0.00019375 C -0.000863125 -0.0998031 -1.096566 -0.996288 -1.196176 -1.593944 " transform="matrix(2,0,0,-2,84.12282,70.8668)"></path> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.196201 1.593835 C -1.096591 0.996179 -0.000888125 0.0996944 0.29794 0.000085 C -0.000888125 -0.0995244 -1.096591 -0.996009 -1.196201 -1.593665 " transform="matrix(0,-2,-2,0,155.90642,71.78338)"></path> </g> </svg> </div> <p>The longitude of the trefoil with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and this orientation is then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mi>a</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">c a b</annotation></semantics></math>.</p> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>Consider the <a class="existingWikiWord" href="/nlab/show/Hopf+link">Hopf link</a>, in which both components have been equipped with a chosen point and an orientation as shown below. Labels have been chosen for the arcs.</p> <div style="text-align: center"> <svg xmlns:xlink="http://www.w3.org/1999/xlink" width="113.386pt" height="113.386pt" viewBox="0 0 113.386 113.386" version="1.1" xmlns="http://www.w3.org/2000/svg"> <defs> <g> <symbol overflow="visible" id="822974523-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="822974523-glyph0-1"> <path style="stroke:none;" d="M 3.71875 -3.765625 C 3.53125 -4.140625 3.25 -4.40625 2.796875 -4.40625 C 1.640625 -4.40625 0.40625 -2.9375 0.40625 -1.484375 C 0.40625 -0.546875 0.953125 0.109375 1.71875 0.109375 C 1.921875 0.109375 2.421875 0.0625 3.015625 -0.640625 C 3.09375 -0.21875 3.453125 0.109375 3.921875 0.109375 C 4.28125 0.109375 4.5 -0.125 4.671875 -0.4375 C 4.828125 -0.796875 4.96875 -1.40625 4.96875 -1.421875 C 4.96875 -1.53125 4.875 -1.53125 4.84375 -1.53125 C 4.75 -1.53125 4.734375 -1.484375 4.703125 -1.34375 C 4.53125 -0.703125 4.359375 -0.109375 3.953125 -0.109375 C 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28.348125 C -1.992656 27.246562 -1.102031 26.355937 -0.00046875 26.355937 C 1.101094 26.355937 1.991719 27.246562 1.991719 28.348125 Z M 1.991719 28.348125 " transform="matrix(1,0,0,-1,42.52,42.52)"></path> <path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 58.687031 -28.347188 C 58.687031 -27.245625 57.7925 -26.355 56.694844 -26.355 C 55.593281 -26.355 54.702656 -27.245625 54.702656 -28.347188 C 54.702656 -29.44875 55.593281 -30.339375 56.694844 -30.339375 C 57.7925 -30.339375 58.687031 -29.44875 58.687031 -28.347188 Z M 58.687031 -28.347188 " transform="matrix(1,0,0,-1,42.52,42.52)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#822974523-glyph0-1" x="5.387" y="58.838"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#822974523-glyph0-2" x="68.728" y="109.651"></use> </g> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.195783 1.594413 C -1.096173 0.996756 -0.00047 0.100272 0.298358 0.0006625 C -0.00047 -0.0989469 -1.096173 -0.995431 -1.195783 -1.593087 " transform="matrix(0,-2,-2,0,14.1732,43.43656)"></path> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 1.19481 1.593556 C 1.095201 0.9959 -0.0005025 0.0994156 -0.299331 -0.00019375 C -0.0005025 -0.0998031 1.095201 -0.996288 1.19481 -1.593944 " transform="matrix(2,0,0,-2,58.23538,70.8668)"></path> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 1.195347 1.593984 C 1.095738 0.996328 0.000035 0.0998438 -0.298793 0.000234375 C 0.000035 -0.099375 1.095738 -0.995859 1.195347 -1.593516 " transform="matrix(0,-2,-2,0,42.52,87.75007)"></path> <path style="fill:none;stroke-width:0.31879;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -1.195169 1.593516 C -1.095559 0.995859 0.00014375 0.099375 0.298972 -0.000234375 C 0.00014375 -0.0998437 -1.095559 -0.996328 -1.195169 -1.593984 " transform="matrix(-2,0,0,2,84.53935,42.52)"></path> </g> </svg> </div> <p>The longitude of the component to which the arc <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> belongs is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">b^{-1}</annotation></semantics></math>. The longitude of the component to which the arc <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> belongs is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a^{-1}</annotation></semantics></math>.</p> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>The longitude of a component of an oriented link diagram is unique up to <a class="existingWikiWord" href="/nlab/show/rotation+permutation">rotation permutation</a>: the longitude obtained using any one choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a rotation permutation of the longitude obtained using any other choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> <p>When it comes to 3-manifolds, for example when describing the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, everything is typically invariant under rotation permutation. Thus it is usual to speak of <em>the</em> longitude of a component of an oriented link diagram.</p> </div> </p> <div class="property">category: <a class="category_link" href="/nlab/all_pages/knot+theory">knot theory</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on January 18, 2019 at 13:21:20. 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