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<div class="ChapSects"><a href="chap32.html#X83856E7178651841">32 <span class="Heading"> Knots and Quandles </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7CFDEEC07F15CF82">32.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X8110BAD17D13F62D">32.1-1 PresentationKnotQuandle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7E82CBA08724AEAA">32.1-2 PD2GC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X81B7CD81869D5583">32.1-3 PlanarDiagramKnot</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7E9458058084E240">32.1-4 GaussCodeKnot</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X8012B9B17BD20990">32.1-5 PresentationKnotQuandleKnot</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X83DE5BA878103191">32.1-6 NumberOfHomomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X84A706527FE23BEB">32.1-7 PartitionedNumberOfHomomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X82013FC97875ADBC">32.1-8 ConjugationQuandle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7F14F7B478D2BEB9">32.1-9 FirstQuandleAxiomIsSatisfied</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7CD0A53778B4B316">32.1-10 IsQuandle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7A57441D7B508D15">32.1-11 Quandles</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X87EF4BF57864D642">32.1-12 Quandle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7BB746B478EC8B5F">32.1-13 IdQuandle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7AAECD4B7A1EA8A3">32.1-14 IsLatin</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X876784FB7A4F28AF">32.1-15 IsConnectedQuandle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X78C1102681E84FDC">32.1-16 ConnectedQuandles</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7C27E982797F6B08">32.1-17 ConnectedQuandle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X841BBAA87A1710E6">32.1-18 IdConnectedQuandle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X82D6ECA279C543B9">32.1-19 IsQuandleEnvelope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X87C70FD17E57A4C5">32.1-20 QuandleQuandleEnvelope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7DAE45E17A191E6E">32.1-21 KnotInvariantCedric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X82F7689578D5EBAD">32.1-22 RightMultiplicationGroupAsPerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X85EA01BD7F66DE1B">32.1-23 RightMultiplicationGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7C9885CC825835FC">32.1-24 AutomorphismGroupQuandleAsPerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap32.html#X7B3890EB85EB004A">32.1-25 AutomorphismGroupQuandle</a></span>
</div></div>
</div>
<h3>32 <span class="Heading"> Knots and Quandles </span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>32.1 <span class="Heading"> </span></h4>
<p>Knots</p>
<p><a id="X8110BAD17D13F62D" name="X8110BAD17D13F62D"></a></p>
<h5>32.1-1 PresentationKnotQuandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationKnotQuandle</code>( <var class="Arg">gaussCode</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Gauss Code of a knot (with the orientations; see <span class="SimpleMath">GaussCodeOfPureCubicalKnot</span> in HAP package) and outputs the generators and relators of the knot quandle associated (in the form of a record).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">4</a></span> </p>
<p><a id="X7E82CBA08724AEAA" name="X7E82CBA08724AEAA"></a></p>
<h5>32.1-2 PD2GC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PD2GC</code>( <var class="Arg">PD</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Planar Diagram of a knot; outputs the Gauss Code associated (with the orientations).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">3</a></span> </p>
<p><a id="X81B7CD81869D5583" name="X81B7CD81869D5583"></a></p>
<h5>32.1-3 PlanarDiagramKnot</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PlanarDiagramKnot</code>( <var class="Arg">n</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns a Planar Diagram for the <span class="SimpleMath">k</span>-th knot with <span class="SimpleMath">n</span> crossings (<span class="SimpleMath">n ≤ 12</span>) if it exists; fail otherwise.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">3</a></span> </p>
<p><a id="X7E9458058084E240" name="X7E9458058084E240"></a></p>
<h5>32.1-4 GaussCodeKnot</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GaussCodeKnot</code>( <var class="Arg">n</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns a Gauss Code (with orientations) for the <span class="SimpleMath">k</span>-th knot with <span class="SimpleMath">n</span> crossings (<span class="SimpleMath">n ≤ 12</span>) if it exists; fail otherwise.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8012B9B17BD20990" name="X8012B9B17BD20990"></a></p>
<h5>32.1-5 PresentationKnotQuandleKnot</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationKnotQuandleKnot</code>( <var class="Arg">n</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns generators and relators (in the form of a record) for the <span class="SimpleMath">k</span>-th knot with <span class="SimpleMath">n</span> crossings (<span class="SimpleMath">n ≤ 12</span>) if it exists; fail otherwise.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">3</a></span> </p>
<p><a id="X83DE5BA878103191" name="X83DE5BA878103191"></a></p>
<h5>32.1-6 NumberOfHomomorphisms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NumberOfHomomorphisms</code>( <var class="Arg">genRelQ</var>, <var class="Arg">finiteQ</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs generators and relators <span class="SimpleMath">genRelQ</span> of a knot quandle (in the form of a record, see above) and a finite quandle <span class="SimpleMath">finiteQ</span>; outputs the number of homomorphisms from the former to the latter.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">3</a></span> </p>
<p><a id="X84A706527FE23BEB" name="X84A706527FE23BEB"></a></p>
<h5>32.1-7 PartitionedNumberOfHomomorphisms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PartitionedNumberOfHomomorphisms</code>( <var class="Arg">genRelQ</var>, <var class="Arg">finiteQ</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs generators and relators <span class="SimpleMath">genRelQ</span> of a knot quandle (in the form of a record, see above) and a finite connected quandle <span class="SimpleMath">finiteQ</span>; outputs a partition of the number of homomorphisms from the former to the latter.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">1</a></span> </p>
<p>Quandles</p>
<p><a id="X82013FC97875ADBC" name="X82013FC97875ADBC"></a></p>
<h5>32.1-8 ConjugationQuandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ConjugationQuandle</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">G</span> and an integer <span class="SimpleMath">n</span>; outputs the associated <span class="SimpleMath">n</span>-fold conjugation quandle.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> </p>
<p><a id="X7F14F7B478D2BEB9" name="X7F14F7B478D2BEB9"></a></p>
<h5>32.1-9 FirstQuandleAxiomIsSatisfied</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FirstQuandleAxiomIsSatisfied</code>( <var class="Arg">M</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">&#an style='color: green'>8227; SecondQuandleAxiomIsSatisfied</code>( <var class="Arg">M</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">&#an style='color: green'>8227; ThirdQuandleAxiomIsSatisfied</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite magma <span class="SimpleMath">M</span>; returns true if <span class="SimpleMath">M</span> satisfy the first/second/third axiom of a quandle, false otherwise.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7CD0A53778B4B316" name="X7CD0A53778B4B316"></a></p>
<h5>32.1-10 IsQuandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsQuandle</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite magma <span class="SimpleMath">M</span>; returns true if <span class="SimpleMath">M</span> is a quandle, false otherwise.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">3</a></span> </p>
<p><a id="X7A57441D7B508D15" name="X7A57441D7B508D15"></a></p>
<h5>32.1-11 Quandles</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Quandles</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns a list of all quandles of size <span class="SimpleMath">n</span>, <span class="SimpleMath">n ≤ 6</span>. If <span class="SimpleMath">n ≥ 7</span>, it returns fail.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">6</a></span> </p>
<p><a id="X87EF4BF57864D642" name="X87EF4BF57864D642"></a></p>
<h5>32.1-12 Quandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Quandle</code>( <var class="Arg">n</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the <span class="SimpleMath">k</span>-th quandle of size <span class="SimpleMath">n</span> (<span class="SimpleMath">n ≤ 6</span>) if such a quandle exists, fail otherwise.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.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/aboutCoveringSpaces.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/aboutQuandles2.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">7</a></span> </p>
<p><a id="X7BB746B478EC8B5F" name="X7BB746B478EC8B5F"></a></p>
<h5>32.1-13 IdQuandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IdQuandle</code>( <var class="Arg">Q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a quandle <span class="SimpleMath">Q</span>; and outputs a list of integers [<span class="SimpleMath">n</span>,<span class="SimpleMath">k</span>] such that <span class="SimpleMath">Q</span> is isomorphic to <span class="SimpleMath">Quandle(n,k)</span>. If <span class="SimpleMath">n ≥ 7</span>, then it returns [<span class="SimpleMath">n</span>,fail] (where <span class="SimpleMath">n</span> is the size of <span class="SimpleMath">Q</span>).</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7AAECD4B7A1EA8A3" name="X7AAECD4B7A1EA8A3"></a></p>
<h5>32.1-14 IsLatin</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsLatin</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a finite quandle <span class="SimpleMath">Q</span>; returns true if <span class="SimpleMath">Q</span> is latin, false otherwise.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X876784FB7A4F28AF" name="X876784FB7A4F28AF"></a></p>
<h5>32.1-15 IsConnectedQuandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsConnectedQuandle</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a finite quandle <span class="SimpleMath">Q</span>; returns true if <span class="SimpleMath">Q</span> is connected, false otherwise.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X78C1102681E84FDC" name="X78C1102681E84FDC"></a></p>
<h5>32.1-16 ConnectedQuandles</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ConnectedQuandles</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns a list of all connected quandles of size <span class="SimpleMath">n</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">3</a></span> </p>
<p><a id="X7C27E982797F6B08" name="X7C27E982797F6B08"></a></p>
<h5>32.1-17 ConnectedQuandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ConnectedQuandle</code>( <var class="Arg">n</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the <span class="SimpleMath">k</span>-th quandle of size <span class="SimpleMath">n</span> if such a quandle exists, fail otherwise.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">4</a></span> </p>
<p><a id="X841BBAA87A1710E6" name="X841BBAA87A1710E6"></a></p>
<h5>32.1-18 IdConnectedQuandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IdConnectedQuandle</code>( <var class="Arg">Q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a connected quandle <span class="SimpleMath">Q</span>; and outputs a list of integers [<span class="SimpleMath">n</span>,<span class="SimpleMath">k</span>] such that <span class="SimpleMath">Q</span> is isomorphic to <span class="SimpleMath">ConnectedQuandle(n,k)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">1</a></span> </p>
<p><a id="X82D6ECA279C543B9" name="X82D6ECA279C543B9"></a></p>
<h5>32.1-19 IsQuandleEnvelope</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsQuandleEnvelope</code>( <var class="Arg">Q</var>, <var class="Arg">G</var>, <var class="Arg">e</var>, <var class="Arg">stigma</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a set <span class="SimpleMath">Q</span>, a permutation group <span class="SimpleMath">G</span>, an element <span class="SimpleMath">e ∈ Q</span> and an element <span class="SimpleMath">stigma ∈ G</span>; returns true if this structure describes a quandle envelope, false otherwise.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">3</a></span> </p>
<p><a id="X87C70FD17E57A4C5" name="X87C70FD17E57A4C5"></a></p>
<h5>32.1-20 QuandleQuandleEnvelope</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; QuandleQuandleEnvelope</code>( <var class="Arg">Q</var>, <var class="Arg">G</var>, <var class="Arg">e</var>, <var class="Arg">stigma</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a set <span class="SimpleMath">Q</span>, a permutation group <span class="SimpleMath">G</span>, an element <span class="SimpleMath">e ∈ Q</span> and an element <span class="SimpleMath">stigma ∈ G</span>. If this structure describes a quandle envelope, the function returns the quandle from this quandle envelope; and fail otherwise. Nb: this quandle is a connected quandle.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">3</a></span> </p>
<p><a id="X7DAE45E17A191E6E" name="X7DAE45E17A191E6E"></a></p>
<h5>32.1-21 KnotInvariantCedric</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; KnotInvariantCedric</code>( <var class="Arg">genRelQ</var>, <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs generators and relators of a knot quandle (in the form of a record, see above) and two integers <span class="SimpleMath">n</span> and <span class="SimpleMath">m</span>; outputs a list [<span class="SimpleMath">n</span>1,<span class="SimpleMath">n</span>2,...,<span class="SimpleMath">n</span>k] where <span class="SimpleMath">n</span>j is a partition of the number of homomorphisms from the considered knot quandle to the <span class="SimpleMath">j</span>-th connected quandle of size <span class="SimpleMath">n ≤ i ≤ m</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X82F7689578D5EBAD" name="X82F7689578D5EBAD"></a></p>
<h5>32.1-22 RightMultiplicationGroupAsPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RightMultiplicationGroupAsPerm</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a connected quandle <span class="SimpleMath">Q</span>; output its right multiplication group whose elements are permutations.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X85EA01BD7F66DE1B" name="X85EA01BD7F66DE1B"></a></p>
<h5>32.1-23 RightMultiplicationGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RightMultiplicationGroup</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a connected quandle <span class="SimpleMath">Q</span>; output its right multiplication group whose elements are mappings from <span class="SimpleMath">Q</span> to <span class="SimpleMath">Q</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7C9885CC825835FC" name="X7C9885CC825835FC"></a></p>
<h5>32.1-24 AutomorphismGroupQuandleAsPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AutomorphismGroupQuandleAsPerm</code>( <var class="Arg">Q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a connected quandle <span class="SimpleMath">Q</span>; outputs its automorphism group whose elements are permutations.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7B3890EB85EB004A" name="X7B3890EB85EB004A"></a></p>
<h5>32.1-25 AutomorphismGroupQuandle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AutomorphismGroupQuandle</code>( <var class="Arg">Q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a connected quandle <span class="SimpleMath">Q</span>; outputs its automorphism group whose elements are mappings from <span class="SimpleMath">Q</span> to <span class="SimpleMath">Q</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutQuandles2.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnotsQuandles.html">3</a></span> </p>
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