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<div class="ChapSects"><a href="chap31_mj.html#X82DADC508677F1EE">31 <span class="Heading"> Knots and Links</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31_mj.html#X7CFDEEC07F15CF82">31.1 <span class="Heading">  </span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31_mj.html#X869065F77C4761EC">31.1-1 PureCubicalKnot</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31_mj.html#X7C7B70C2788884AC">31.1-2 ViewPureCubicalKnot</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31_mj.html#X7D86D13C822D59A9">31.1-3 KnotSum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31_mj.html#X797F8D4A848DD9BC">31.1-4 KnotGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31_mj.html#X7A87E5EB82E67589">31.1-5 AlexanderMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31_mj.html#X7DC474EE7A909563">31.1-6 AlexanderPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31_mj.html#X853774F885B17523">31.1-7 ProjectionOfPureCubicalComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31_mj.html#X7D8681B079E019C0">31.1-8 ReadPDBfileAsPureCubicalComplex</a></span>
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<h3>31 <span class="Heading"> Knots and Links</span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>31.1 <span class="Heading">  </span></h4>

<p><a id="X869065F77C4761EC" name="X869065F77C4761EC"></a></p>

<h5>31.1-1 PureCubicalKnot</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PureCubicalKnot</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PureCubicalKnot</code>( <var class="Arg">n</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(L=[[m1,n1], [m2,n2], ..., [mk,nk]]\)</span> of pairs of integers describing a cubical arc presentation of a link with all vertical lines at the front and all horizontal lines at the back. The bottom horizontal line extends from the m1-th column to the n1-th column. The second to bottom horizontal line extends from the m2-th column to the n2-th column. And so on. The link is returned as a 3-dimensional pure cubical complex.</p>

<p>Alternatively the function inputs two integers <span class="SimpleMath">\(n\)</span>, <span class="SimpleMath">\(i\)</span> and returns the <span class="SimpleMath">\(i\)</span>-th prime knot on <span class="SimpleMath">\(n\)</span> crossings.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap2.html">2</a></span> , <span class="URL"><a href="../tutorial/chap3.html">3</a></span> , <span class="URL"><a href="../tutorial/chap4.html">4</a></span> , <span class="URL"><a href="../tutorial/chap6.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">11</a></span> </p>

<p><a id="X7C7B70C2788884AC" name="X7C7B70C2788884AC"></a></p>

<h5>31.1-2 ViewPureCubicalKnot</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ViewPureCubicalKnot</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical link <span class="SimpleMath">\(L\)</span> and displays it.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">2</a></span> </p>

<p><a id="X7D86D13C822D59A9" name="X7D86D13C822D59A9"></a></p>

<h5>31.1-3 KnotSum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KnotSum</code>( <var class="Arg">K</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two pure cubical knots <span class="SimpleMath">\(K\)</span>, <span class="SimpleMath">\(L\)</span> and returns their sum as a pure cubical knot. This function is not defined for links with more than one component.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap2.html">1</a></span> , <span class="URL"><a href="../tutorial/chap3.html">2</a></span> , <span class="URL"><a href="../tutorial/chap6.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">5</a></span> </p>

<p><a id="X797F8D4A848DD9BC" name="X797F8D4A848DD9BC"></a></p>

<h5>31.1-4 KnotGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KnotGroup</code>( <var class="Arg">K</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical link <span class="SimpleMath">\(K\)</span> and returns the fundamental group of its complement. The group is returned as a finitely presented group.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">1</a></span> </p>

<p><a id="X7A87E5EB82E67589" name="X7A87E5EB82E67589"></a></p>

<h5>31.1-5 AlexanderMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlexanderMatrix</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finitely presented group <span class="SimpleMath">\(G\)</span> whose abelianization is infinite cyclic. It returns the Alexander matrix of the presentation.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7DC474EE7A909563" name="X7DC474EE7A909563"></a></p>

<h5>31.1-6 AlexanderPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlexanderPolynomial</code>( <var class="Arg">K</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlexanderPolynomial</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a pure cubical knot <span class="SimpleMath">\(K\)</span> or a finitely presented group <span class="SimpleMath">\(G\)</span> whose abelianization is infinite cyclic. The Alexander Polynomial is returned.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap2.html">2</a></span> , <span class="URL"><a href="../tutorial/chap5.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">4</a></span> </p>

<p><a id="X853774F885B17523" name="X853774F885B17523"></a></p>

<h5>31.1-7 ProjectionOfPureCubicalComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectionOfPureCubicalComplex</code>( <var class="Arg">K</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an $n$-dimensional pure cubical complex <span class="SimpleMath">\(K\)</span> and returns an n-1-dimensional pure cubical complex K'. The returned complex is obtained by projecting Euclidean n-space onto Euclidean n-1-space.



<p><strong class="button">Examples:</strong></p>

<p><a id="X7D8681B079E019C0" name="X7D8681B079E019C0"></a></p>

<h5>31.1-8 ReadPDBfileAsPureCubicalComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadPDBfileAsPureCubicalComplex</code>( <var class="Arg">file</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadPDBfileAsPureCubicalComplex</code>( <var class="Arg">file</var>, <var class="Arg">m</var>, <var class="Arg">c</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a protein database file describing a protein, and optionally inputs a positive integer m and character string c. The default values for the optional inputs are m=5 and c="A". It loads the chain of amino acids labelled by c in the file as a 3-dimensional pure cubical complex of the homotopy type of a circle.</p>

<p>It might happen that the function fails to construct a pure cubical complex of the homotopy type of a circle. In this case retry with a larger integer m.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">3</a></span> </p>


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