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<div class="ChapSects"><a href="chap1_mj.html#X85BEB9F48106583E">1 <span class="Heading">Basic functionality for cellular complexes, fundamental groups and homology</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7F06418383E098EB">1.1 <span class="Heading"> Data <span class="SimpleMath">\(\longrightarrow\)</span> Cellular Complexes </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85C818B87D9AC922">1.1-1 RegularCWPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7910F39B7AB79096">1.1-2 CubicalComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78A3981C878C7FB5">1.1-3 PureCubicalComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X869065F77C4761EC">1.1-4 PureCubicalKnot</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B432A6184CBAC75">1.1-5 PurePermutahedralKnot</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X824625A27FF6DE6F">1.1-6 PurePermutahedralComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80CAD0357AF44E48">1.1-7 CayleyGraphOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8187F6507BA14D5C">1.1-8 EquivariantEuclideanSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FE0522B8134DF7C">1.1-9 EquivariantOrbitPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X81E8E97278B1AE92">1.1-10 EquivariantTwoComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F8D4C4C7ED15A31">1.1-11 QuillenComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X854B96757AF38A41">1.1-12 RestrictedEquivariantCWComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A3B6B647C8CF90B">1.1-13 RandomSimplicialGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8394037487D3C17E">1.1-14 RandomSimplicialTwoComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83DB403087D02CC8">1.1-15 ReadCSVfileAsPureCubicalKnot</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BE9892784AA4990">1.1-16 ReadImageAsPureCubicalComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84D89B96873308B7">1.1-17 ReadImageAsFilteredPureCubicalComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80E8B89F7E95D101">1.1-18 ReadImageAsWeightFunction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D8681B079E019C0">1.1-19 ReadPDBfileAsPureCubicalComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E278788808A9EE4">1.1-20 ReadPDBfileAsPurepermutahedralComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85C818B87D9AC922">1.1-21 RegularCWPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X818F2E887FE5F7BE">1.1-22 SimplicialComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79CA51F27C07435C">1.1-23 SymmetricMatrixToFilteredGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8227636B7E878448">1.1-24 SymmetricMatrixToGraph</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7C0C080487641830">1.2 <span class="Heading"> Metric Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F8113757F7DD2F4">1.2-1 CayleyMetric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A4560307BA911F5">1.2-2 EuclideanMetric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X789AE7CE8445A67C">1.2-3 EuclideanSquaredMetric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79DA33CB7D46CAB4">1.2-4 HammingMetric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BD62D75829F8701">1.2-5 KendallMetric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8763D1167EF519A1">1.2-6 ManhattanMetric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C86B58A7CEA5513">1.2-7 VectorsToSymmetricMatrix</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X80A49CAC84313990">1.3 <span class="Heading"> Cellular Complexes <span class="SimpleMath">\(\longrightarrow\)</span> Cellular Complexes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7AF313D387F6BA22">1.3-1 BoundaryMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X848ED6C378A1C5C0">1.3-2 CliqueComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85FAD5E086DBD429">1.3-3 ConcentricFiltration</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X861BA02C7902A4F4">1.3-4 DirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DB4D3B57E0DA723">1.3-5 FiltrationTerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B335342839E5146">1.3-6 Graph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7966519E78BC6C18">1.3-7 HomotopyGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84560FF678621AE1">1.3-8 Nerve</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C2BEF7C871E54D7">1.3-9 RegularCWComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79967AC2859A9631">1.3-10 RegularCWMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82843E747FE622AF">1.3-11 ThickeningFiltration</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7FD50DF6782F00A0">1.4 <span class="Heading"> Cellular Complexes <span class="SimpleMath">\(\longrightarrow\)</span> Cellular Complexes (Preserving Data Types)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X840576107A2907B8">1.4-1 ContractedComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A46614B84FF25BE">1.4-2 ContractibleSubcomplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86164F4481ACC485">1.4-3 KnotReflection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D86D13C822D59A9">1.4-4 KnotSum</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X855537287E9C4E72">1.4-5 OrientRegularCWComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A266B5A7BE88E89">1.4-6 PathComponent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FF34B9E86E901DC">1.4-7 PureComplexBoundary</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D0C9B27845F0739">1.4-8 PureComplexComplement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FB5BE6C78D5C7C8">1.4-9 PureComplexDifference</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8091C9BA819C2332">1.4-10 PureComplexInterstection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84A7E7A47F7BA09D">1.4-11 PureComplexThickened</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78014E027F28C2C8">1.4-12 PureComplexUnion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E7AC0E77E25C45B">1.4-13 SimplifiedComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X844174D37E70B9B4">1.4-14 ZigZagContractedComplex</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7E25932F7DD535E8">1.5 <span class="Heading"> Cellular Complexes <span class="SimpleMath">\(\longrightarrow\)</span> Homotopy Invariants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DC474EE7A909563">1.5-1 AlexanderPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83EF7B888014C363">1.5-2 BettiNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8307F8DB85F145AE">1.5-3 EulerCharacteristic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78813B9A851B922A">1.5-4 EulerIntegral</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EAE7E4181546C17">1.5-5 FundamentalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X808733FF7EF6278E">1.5-6 FundamentalGroupOfQuotient</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78F2C5ED80D1C8DD">1.5-7 IsAspherical</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X797F8D4A848DD9BC">1.5-8 KnotGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X825539B57FBDDE86">1.5-9 PiZero</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EE96E8B7C1643BD">1.5-10 PersistentBettiNumbers</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7C17A7897DDAE22C">1.6 <span class="Heading"> Data <span class="SimpleMath">\(\longrightarrow\)</span> Homotopy Invariants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F5B6CAD7CB2E985">1.6-1 DendrogramMat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X859286BF7F6047B7">1.7 <span class="Heading"> Cellular Complexes <span class="SimpleMath">\(\longrightarrow\)</span> Non Homotopy Invariants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A1C427578108B7E">1.7-1 ChainComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D4AF2E8785DA457">1.7-2 ChainComplexEquivalence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D77D18679E941D3">1.7-3 ChainComplexOfQuotient</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BCD94877DF261C4">1.7-4 ChainMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B8741FB7A3263EC">1.7-5 CochainComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8489A39F870FF08B">1.7-6 CriticalCells</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A4AD52D82627ABC">1.7-7 DiagonalApproximation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X858ADA3B7A684421">1.7-8 Size</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7B6F366F7A2D8FEE">1.8 <span class="Heading"> (Co)chain Complexes <span class="SimpleMath">\(\longrightarrow \)</span> (Co)chain Complexes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X829DD3868410FE2E">1.8-1 FilteredTensorWithIntegers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BC291C47FEAC5B8">1.8-2 FilteredTensorWithIntegersModP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X788F3B5E7810E309">1.8-3 HomToIntegers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8122D25786C83565">1.8-4 TensorWithIntegersModP</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7BB8DC9783A4AF81">1.9 <span class="Heading"> (Co)chain Complexes <span class="SimpleMath">\(\longrightarrow \)</span> Homotopy Invariants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84CFC57B7E9CCCF7">1.9-1 Cohomology</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X877825E57D79839C">1.9-2 CupProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85A9D5CB8605329C">1.9-3 Homology</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X867BE1388467C939">1.10 <span class="Heading"> Visualization</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X806A81EF79CE0DEF">1.10-1 BarCodeDisplay</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83D60A6682EBB6F1">1.10-2 BarCodeCompactDisplay</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80CAD0357AF44E48">1.10-3 CayleyGraphOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83A5C59278E13248">1.10-4 Display</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B98A3C4831D5B0D">1.10-5 DisplayArcPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X861690C27BADC326">1.10-6 DisplayCSVKnotFile</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F4AA01E7C0A5C16">1.10-7 DisplayDendrogram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E5A38F081B401BE">1.10-8 DisplayDendrogramMat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X822F54F385D7EF8A">1.10-9 DisplayPDBfile</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80EC50C27EFF2E12">1.10-10 OrbitPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DF49EAD7C0B0E84">1.10-11 ScatterPlot</a></span>
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</div>
<h3>1 <span class="Heading">Basic functionality for cellular complexes, fundamental groups and homology</span></h3>
<p>This page covers the functions used in chapters 1 and 2 of the book <span class="URL"><a href="https://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832980">An Invitation to Computational Homotopy</a></span>.</p>
<p><a id="X7F06418383E098EB" name="X7F06418383E098EB"></a></p>
<h4>1.1 <span class="Heading"> Data <span class="SimpleMath">\(\longrightarrow\)</span> Cellular Complexes </span></h4>
<p><a id="X85C818B87D9AC922" name="X85C818B87D9AC922"></a></p>
<h5>1.1-1 RegularCWPolytope</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RegularCWPolytope</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">‣ RegularCWPolytope</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(L\)</span> of vectors in <span class="SimpleMath">\(\mathbb R^n\)</span> and outputs their convex hull as a regular CW-complex.</p>
<p>Inputs a permutation group G of degree <span class="SimpleMath">\(d\)</span> and vector <span class="SimpleMath">\(v\in \mathbb R^d\)</span>, and outputs the convex hull of the orbit <span class="SimpleMath">\(\{v^g : g\in G\}\)</span> as a regular CW-complex.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7910F39B7AB79096" name="X7910F39B7AB79096"></a></p>
<h5>1.1-2 CubicalComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CubicalComplex</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a binary array <span class="SimpleMath">\(A\)</span> and returns the cubical complex represented by <span class="SimpleMath">\(A\)</span>. The array <span class="SimpleMath">\(A\)</span> must of course be such that it represents a cubical complex.</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/chap5.html">3</a></span> , <span class="URL"><a href="../tutorial/chap10.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLinks.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCubical.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRandomComplexes.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTDA.html">11</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">12</a></span> </p>
<p><a id="X78A3981C878C7FB5" name="X78A3981C878C7FB5"></a></p>
<h5>1.1-3 PureCubicalComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PureCubicalComplex</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a binary array <span class="SimpleMath">\(A\)</span> and returns the pure cubical complex represented by <span class="SimpleMath">\(A\)</span>.</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/chap5.html">3</a></span> , <span class="URL"><a href="../tutorial/chap10.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLinks.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCubical.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRandomComplexes.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTDA.html">11</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">12</a></span> </p>
<p><a id="X869065F77C4761EC" name="X869065F77C4761EC"></a></p>
<h5>1.1-4 PureCubicalKnot</h5>
<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">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">‣ PureCubicalKnot</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs integers <span class="SimpleMath">\(n, k\)</span> and returns the <span class="SimpleMath">\(k\)</span>-th prime knot on <span class="SimpleMath">\(n\)</span> crossings as a pure cubical complex (if this prime knot exists).</p>
<p>Inputs a list <span class="SimpleMath">\(L\)</span> describing an arc presentation for a knot or link and returns the knot or link as a pure cubical complex.</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="X7B432A6184CBAC75" name="X7B432A6184CBAC75"></a></p>
<h5>1.1-5 PurePermutahedralKnot</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PurePermutahedralKnot</code>( <var class="Arg">n</var>, <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">‣ PurePermutahedralKnot</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs integers <span class="SimpleMath">\(n, k\)</span> and returns the <span class="SimpleMath">\(k\)</span>-th prime knot on <span class="SimpleMath">\(n\)</span> crossings as a pure permutahedral complex (if this prime knot exists).</p>
<p>Inputs a list <span class="SimpleMath">\(L\)</span> describing an arc presentation for a knot or link and returns the knot or link as a pure permutahedral complex.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> </p>
<p><a id="X824625A27FF6DE6F" name="X824625A27FF6DE6F"></a></p>
<h5>1.1-6 PurePermutahedralComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PurePermutahedralComplex</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a binary array <span class="SimpleMath">\(A\)</span> and returns the pure permutahedral complex represented by <span class="SimpleMath">\(A\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap2.html">1</a></span> , <span class="URL"><a href="../tutorial/chap5.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPeripheral.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCubical.html">4</a></span> </p>
<p><a id="X80CAD0357AF44E48" name="X80CAD0357AF44E48"></a></p>
<h5>1.1-7 CayleyGraphOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CayleyGraphOfGroup</code>( <var class="Arg">G</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">\(G\)</span> and a list <span class="SimpleMath">\(L\)</span> of elements in <span class="SimpleMath">\(G\)</span>.It returns the Cayley graph of the group generated by <span class="SimpleMath">\(L\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8187F6507BA14D5C" name="X8187F6507BA14D5C"></a></p>
<h5>1.1-8 EquivariantEuclideanSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivariantEuclideanSpace</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a crystallographic group <span class="SimpleMath">\(G\)</span> with left action on <span class="SimpleMath">\(\mathbb R^n\)</span> together with a row vector <span class="SimpleMath">\(v \in \mathbb R^n\)</span>. It returns an equivariant regular CW-space corresponding to the Dirichlet-Voronoi tessellation of <span class="SimpleMath">\(\mathbb R^n\)</span> produced from the orbit of <span class="SimpleMath">\(v\)</span> under the action.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> </p>
<p><a id="X7FE0522B8134DF7C" name="X7FE0522B8134DF7C"></a></p>
<h5>1.1-9 EquivariantOrbitPolytope</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivariantOrbitPolytope</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group <span class="SimpleMath">\(G\)</span> of degree <span class="SimpleMath">\(n\)</span> together with a row vector <span class="SimpleMath">\(v \in \mathbb R^n\)</span>. It returns, as an equivariant regular CW-space, the convex hull of the orbit of <span class="SimpleMath">\(v\)</span> under the canonical left action of <span class="SimpleMath">\(G\)</span> on <span class="SimpleMath">\(\mathbb R^n\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X81E8E97278B1AE92" name="X81E8E97278B1AE92"></a></p>
<h5>1.1-10 EquivariantTwoComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivariantTwoComplex</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a suitable group <span class="SimpleMath">\(G\)</span> and returns, as an equivariant regular CW-space, the <span class="SimpleMath">\(2\)</span>-complex associated to some presentation of <span class="SimpleMath">\(G\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> </p>
<p><a id="X7F8D4C4C7ED15A31" name="X7F8D4C4C7ED15A31"></a></p>
<h5>1.1-11 QuillenComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuillenComplex</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">\(G\)</span> and prime <span class="SimpleMath">\(p\)</span>, and returns the simplicial complex arising as the order complex of the poset of elementary abelian <span class="SimpleMath">\(p\)</span>-subgroups of <span class="SimpleMath">\(G\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutBredon.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCubical.html">4</a></span> </p>
<p><a id="X854B96757AF38A41" name="X854B96757AF38A41"></a></p>
<h5>1.1-12 RestrictedEquivariantCWComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedEquivariantCWComplex</code>( <var class="Arg">Y</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(G\)</span>-equivariant regular CW-space Y and a subgroup <span class="SimpleMath">\(H \le G\)</span> for which GAP can find a transversal. It returns the equivariant regular CW-complex obtained by retricting the action to <span class="SimpleMath">\(H\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7A3B6B647C8CF90B" name="X7A3B6B647C8CF90B"></a></p>
<h5>1.1-13 RandomSimplicialGraph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomSimplicialGraph</code>( <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an integer <span class="SimpleMath">\( n \ge 1 \)</span> and positive prime <span class="SimpleMath">\(p\)</span>, and returns an Erdős–Rényi random graph as a <span class="SimpleMath">\(1\)</span>-dimensional simplicial complex. The graph has <span class="SimpleMath">\(n\)</span> vertices. Each pair of vertices is, with probability <span class="SimpleMath">\(p\)</span>, directly connected by an edge.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutRandomComplexes.html">1</a></span> </p>
<p><a id="X8394037487D3C17E" name="X8394037487D3C17E"></a></p>
<h5>1.1-14 RandomSimplicialTwoComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomSimplicialTwoComplex</code>( <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an integer <span class="SimpleMath">\( n \ge 1 \)</span> and positive prime <span class="SimpleMath">\(p\)</span>, and returns a Linial-Meshulam random simplicial <span class="SimpleMath">\(2\)</span>-complex. The <span class="SimpleMath">\(1\)</span>-skeleton of this simplicial complex is the complete graph on <span class="SimpleMath">\(n\)</span> vertices. Each triple of vertices lies, with probability <span class="SimpleMath">\(p\)</span>, in a common <span class="SimpleMath">\(2\)</span>-simplex of the complex.</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/aboutRandomComplexes.html">2</a></span> </p>
<p><a id="X83DB403087D02CC8" name="X83DB403087D02CC8"></a></p>
<h5>1.1-15 ReadCSVfileAsPureCubicalKnot</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadCSVfileAsPureCubicalKnot</code>( <var class="Arg">str</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">‣ ReadCSVfileAsPureCubicalKnot</code>( <var class="Arg">str</var>, <var class="Arg">r</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">‣ ReadCSVfileAsPureCubicalKnot</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">‣ ReadCSVfileAsPureCubicalKnot</code>( <var class="Arg">L</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Reads a CSV file identified by a string str such as "file.pdb" or "path/file.pdb" and returns a <span class="SimpleMath">\(3\)</span>-dimensional pure cubical complex <span class="SimpleMath">\(K\)</span>. Each line of the file should contain the coordinates of a point in <span class="SimpleMath">\(\mathbb R^3\)</span> and the complex <span class="SimpleMath">\(K\)</span> should represent a knot determined by the sequence of points, though the latter is not guaranteed. A useful check in this direction is to test that <span class="SimpleMath">\(K\)</span> has the homotopy type of a circle.</p>
<p>If the test fails then try the function again with an integer <span class="SimpleMath">\(r \ge 2\)</span> entered as the optional second argument. The integer determines the resolution with which the knot is constructed.</p>
<p>The function can also read in a list <span class="SimpleMath">\(L\)</span> of strings identifying CSV files for several knots. In this case a list <span class="SimpleMath">\(R\)</span> of integer resolutions can also be entered. The lists <span class="SimpleMath">\(L\)</span> and <span class="SimpleMath">\(R\)</span> must be of equal length.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap2.html">1</a></span> </p>
<p><a id="X7BE9892784AA4990" name="X7BE9892784AA4990"></a></p>
<h5>1.1-16 ReadImageAsPureCubicalComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadImageAsPureCubicalComplex</code>( <var class="Arg">str</var>, <var class="Arg">t</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Reads an image file identified by a string str such as "file.bmp", "file.eps", "file.jpg", "path/file.png" etc., together with an integer <span class="SimpleMath">\(t\)</span> between <span class="SimpleMath">\(0\)</span> and <span class="SimpleMath">\(765\)</span>. It returns a <span class="SimpleMath">\(2\)</span>-dimensional pure cubical complex corresponding to a black/white version of the image determined by the threshold <span class="SimpleMath">\(t\)</span>. The <span class="SimpleMath">\(2\)</span>-cells of the pure cubical complex correspond to pixels with RGB value <span class="SimpleMath">\(R+G+B \le t\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCubical.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTDA.html">5</a></span> </p>
<p><a id="X84D89B96873308B7" name="X84D89B96873308B7"></a></p>
<h5>1.1-17 ReadImageAsFilteredPureCubicalComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadImageAsFilteredPureCubicalComplex</code>( <var class="Arg">str</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Reads an image file identified by a string str such as "file.bmp", "file.eps", "file.jpg", "path/file.png" etc., together with a positive integer <span class="SimpleMath">\(n\)</span>. It returns a <span class="SimpleMath">\(2\)</span>-dimensional filtered pure cubical complex of filtration length <span class="SimpleMath">\(n\)</span>. The <span class="SimpleMath">\(k\)</span>th term in the filtration is a pure cubical complex corresponding to a black/white version of the image determined by the threshold <span class="SimpleMath">\(t_k=k \times 765/n \)</span>. The <span class="SimpleMath">\(2\)</span>-cells of the <span class="SimpleMath">\(k\)</span>th term correspond to pixels with RGB value <span class="SimpleMath">\(R+G+B \le t_k\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> </p>
<p><a id="X80E8B89F7E95D101" name="X80E8B89F7E95D101"></a></p>
<h5>1.1-18 ReadImageAsWeightFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadImageAsWeightFunction</code>( <var class="Arg">str</var>, <var class="Arg">t</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Reads an image file identified by a string str such as "file.bmp", "file.eps", "file.jpg", "path/file.png" etc., together with an integer <span class="SimpleMath">\(t\)</span>. It constructs a <span class="SimpleMath">\(2\)</span>-dimensional regular CW-complex <span class="SimpleMath">\(Y\)</span> from the image, together with a weight function <span class="SimpleMath">\(w\colon Y\rightarrow \mathbb Z\)</span> corresponding to a filtration on <span class="SimpleMath">\(Y\)</span> of filtration length <span class="SimpleMath">\(t\)</span>. The pair <span class="SimpleMath">\([Y,w]\)</span> is returned.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7D8681B079E019C0" name="X7D8681B079E019C0"></a></p>
<h5>1.1-19 ReadPDBfileAsPureCubicalComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadPDBfileAsPureCubicalComplex</code>( <var class="Arg">str</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">str</var>, <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Reads a PDB (Protein Database) file identified by a string str such as "file.pdb" or "path/file.pdb" and returns a <span class="SimpleMath">\(3\)</span>-dimensional pure cubical complex <span class="SimpleMath">\(K\)</span>. The complex <span class="SimpleMath">\(K\)</span> should represent a (protein backbone) knot but this is not guaranteed. A useful check in this direction is to test that <span class="SimpleMath">\(K\)</span> has the homotopy type of a circle.</p>
<p>If the test fails then try the function again with an integer <span class="SimpleMath">\(r \ge 2\)</span> entered as the optional second argument. The integer determines the resolution with which the knot is constructed.</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>
<p><a id="X7E278788808A9EE4" name="X7E278788808A9EE4"></a></p>
<h5>1.1-20 ReadPDBfileAsPurepermutahedralComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadPDBfileAsPurepermutahedralComplex</code></td><td class="tdright">( global variable )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadPDBfileAsPurePermutahedralComplex</code>( <var class="Arg">str</var>, <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Reads a PDB (Protein Database) file identified by a string str such as "file.pdb" or "path/file.pdb" and returns a <span class="SimpleMath">\(3\)</span>-dimensional pure permutahedral complex <span class="SimpleMath">\(K\)</span>. The complex <span class="SimpleMath">\(K\)</span> should represent a (protein backbone) knot but this is not guaranteed. A useful check in this direction is to test that <span class="SimpleMath">\(K\)</span> has the homotopy type of a circle.</p>
<p>If the test fails then try the function again with an integer <span class="SimpleMath">\(r \ge 2\)</span> entered as the optional second argument. The integer determines the resolution with which the knot is constructed.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X85C818B87D9AC922" name="X85C818B87D9AC922"></a></p>
<h5>1.1-21 RegularCWPolytope</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RegularCWPolytope</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">‣ RegularCWPolytope</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(L\)</span> of vectors in <span class="SimpleMath">\(\mathbb R^n\)</span> and outputs their convex hull as a regular CW-complex.</p>
<p>Inputs a permutation group G of degree <span class="SimpleMath">\(d\)</span> and vector <span class="SimpleMath">\(v\in \mathbb R^d\)</span>, and outputs the convex hull of the orbit <span class="SimpleMath">\(\{v^g : g\in G\}\)</span> as a regular CW-complex.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X818F2E887FE5F7BE" name="X818F2E887FE5F7BE"></a></p>
<h5>1.1-22 SimplicialComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimplicialComplex</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(L\)</span> whose entries are lists of vertices representing the maximal simplices of a simplicial complex, and returns the simplicial complex. Here a "vertex" is a GAP object such as an integer or a subgroup. The list <span class="SimpleMath">\(L\)</span> can also contain non-maximal simplices.</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/chap5.html">5</a></span> , <span class="URL"><a href="../tutorial/chap10.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCubical.html">11</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRandomComplexes.html">12</a></span> </p>
<p><a id="X79CA51F27C07435C" name="X79CA51F27C07435C"></a></p>
<h5>1.1-23 SymmetricMatrixToFilteredGraph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricMatrixToFilteredGraph</code>( <var class="Arg">A</var>, <var class="Arg">m</var>, <var class="Arg">s</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">‣ SymmetricMatrixToFilteredGraph</code>( <var class="Arg">A</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(n \times n\)</span> symmetric matrix <span class="SimpleMath">\(A\)</span>, a positive integer <span class="SimpleMath">\(m\)</span> and a positive rational <span class="SimpleMath">\(s\)</span>. The function returns a filtered graph of filtration length <span class="SimpleMath">\(m\)</span>. The <span class="SimpleMath">\(t\)</span>-th term of the filtration is a graph with <span class="SimpleMath">\(n\)</span> vertices and an edge between the <span class="SimpleMath">\(i\)</span>-th and <span class="SimpleMath">\(j\)</span>-th vertices if the <span class="SimpleMath">\((i,j)\)</span> entry of <span class="SimpleMath">\(A\)</span> is less than or equal to <span class="SimpleMath">\(t \times s/m\)</span>.</p>
<p>If the optional input <span class="SimpleMath">\(s\)</span> is omitted then it is set equal to the largest entry in the matrix <span class="SimpleMath">\(A\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">3</a></span> </p>
<p><a id="X8227636B7E878448" name="X8227636B7E878448"></a></p>
<h5>1.1-24 SymmetricMatrixToGraph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricMatrixToGraph</code>( <var class="Arg">A</var>, <var class="Arg">t</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(n\times n\)</span> symmetric matrix <span class="SimpleMath">\(A\)</span> over the rationals and a rational number <span class="SimpleMath">\(t \ge 0\)</span>, and returns the graph on the vertices <span class="SimpleMath">\(1,2, \ldots, n\)</span> with an edge between distinct vertices <span class="SimpleMath">\(i\)</span> and <span class="SimpleMath">\(j\)</span> precisely when the <span class="SimpleMath">\((i,j)\)</span> entry of <span class="SimpleMath">\(A\)</span> is <span class="SimpleMath">\(\le t\)</span>.</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/aboutMetrics.html">2</a></span> </p>
<p><a id="X7C0C080487641830" name="X7C0C080487641830"></a></p>
<h4>1.2 <span class="Heading"> Metric Spaces</span></h4>
<p><a id="X7F8113757F7DD2F4" name="X7F8113757F7DD2F4"></a></p>
<h5>1.2-1 CayleyMetric</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CayleyMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two permutations <span class="SimpleMath">\(g,h\)</span> and optionally the degree <span class="SimpleMath">\(N\)</span> of a symmetric group containing them. It returns the minimum number of transpositions needed to express <span class="SimpleMath">\(g*h^{-1}\)</span> as a product of transpositions.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">1</a></span> </p>
<p><a id="X7A4560307BA911F5" name="X7A4560307BA911F5"></a></p>
<h5>1.2-2 EuclideanMetric</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EuclideanMetric</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs two vectors <span class="SimpleMath">\(v,w \in \mathbb R^n\)</span> and returns a rational number approximating the Euclidean distance between them.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X789AE7CE8445A67C" name="X789AE7CE8445A67C"></a></p>
<h5>1.2-3 EuclideanSquaredMetric</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EuclideanSquaredMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two vectors <span class="SimpleMath">\(v,w \in \mathbb R^n\)</span> and returns the square of the Euclidean distance between them.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X79DA33CB7D46CAB4" name="X79DA33CB7D46CAB4"></a></p>
<h5>1.2-4 HammingMetric</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HammingMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two permutations <span class="SimpleMath">\(g,h\)</span> and optionally the degree <span class="SimpleMath">\(N\)</span> of a symmetric group containing them. It returns the minimum number of integers moved by the permutation <span class="SimpleMath">\(g*h^{-1}\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7BD62D75829F8701" name="X7BD62D75829F8701"></a></p>
<h5>1.2-5 KendallMetric</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KendallMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two permutations <span class="SimpleMath">\(g,h\)</span> and optionally the degree <span class="SimpleMath">\(N\)</span> of a symmetric group containing them. It returns the minimum number of adjacent transpositions needed to express <span class="SimpleMath">\(g*h^{-1}\)</span> as a product of adjacent transpositions. An <em>adjacent</em> transposition is of the form <span class="SimpleMath">\((i,i+1)\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8763D1167EF519A1" name="X8763D1167EF519A1"></a></p>
<h5>1.2-6 ManhattanMetric</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ManhattanMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two vectors <span class="SimpleMath">\(v,w \in \mathbb R^n\)</span> and returns the Manhattan distance between them.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">1</a></span> </p>
<p><a id="X7C86B58A7CEA5513" name="X7C86B58A7CEA5513"></a></p>
<h5>1.2-7 VectorsToSymmetricMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorsToSymmetricMatrix</code>( <var class="Arg">V</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">‣ VectorsToSymmetricMatrix</code>( <var class="Arg">V</var>, <var class="Arg">d</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(V =\{ v_1, \ldots, v_k\} \in \mathbb R^n\)</span> and returns the <span class="SimpleMath">\(k \times k\)</span> symmetric matrix of Euclidean distances <span class="SimpleMath">\(d(v_i, v_j)\)</span>. When these distances are irrational they are approximated by a rational number.</p>
<p>As an optional second argument any rational valued function <span class="SimpleMath">\(d(x,y)\)</span> can be entered.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">3</a></span> </p>
<p><a id="X80A49CAC84313990" name="X80A49CAC84313990"></a></p>
<h4>1.3 <span class="Heading"> Cellular Complexes <span class="SimpleMath">\(\longrightarrow\)</span> Cellular Complexes</span></h4>
<p><a id="X7AF313D387F6BA22" name="X7AF313D387F6BA22"></a></p>
<h5>1.3-1 BoundaryMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BoundaryMap</code>( <var class="Arg">K</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure regular CW-complex <span class="SimpleMath">\(K\)</span> and returns the regular CW-inclusion map <span class="SimpleMath">\(\iota \colon \partial K \hookrightarrow K\)</span> from the boundary <span class="SimpleMath">\(\partial K\)</span> into the complex <span class="SimpleMath">\(K\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap2.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTopology.html">3</a></span> </p>
<p><a id="X848ED6C378A1C5C0" name="X848ED6C378A1C5C0"></a></p>
<h5>1.3-2 CliqueComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CliqueComplex</code>( <var class="Arg">G</var>, <var class="Arg">n</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">‣ CliqueComplex</code>( <var class="Arg">F</var>, <var class="Arg">n</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">‣ CliqueComplex</code>( <var class="Arg">K</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph <span class="SimpleMath">\(G\)</span> and integer <span class="SimpleMath">\(n \ge 1\)</span>. It returns the <span class="SimpleMath">\(n\)</span>-skeleton of a simplicial complex <span class="SimpleMath">\(K\)</span> with one <span class="SimpleMath">\(k\)</span>-simplex for each complete subgraph of <span class="SimpleMath">\(G\)</span> on <span class="SimpleMath">\(k+1\)</span> vertices.</p>
<p>Inputs a fitered graph <span class="SimpleMath">\(F\)</span> and integer <span class="SimpleMath">\(n \ge 1\)</span>. It returns the <span class="SimpleMath">\(n\)</span>-skeleton of a filtered simplicial complex <span class="SimpleMath">\(K\)</span> whose <span class="SimpleMath">\(t\)</span>-term has one <span class="SimpleMath">\(k\)</span>-simplex for each complete subgraph of the <span class="SimpleMath">\(t\)</span>-th term of <span class="SimpleMath">\(G\)</span> on <span class="SimpleMath">\(k+1\)</span> vertices.</p>
<p>Inputs a simplicial complex of dimension <span class="SimpleMath">\(d=1\)</span> or <span class="SimpleMath">\(d=2\)</span>. If <span class="SimpleMath">\(d=1\)</span> then the clique complex of a graph returned. If <span class="SimpleMath">\(d=2\)</span> then the clique complex of a <span class="SimpleMath">\(2\)</span>-complex is returned.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> </p>
<p><a id="X85FAD5E086DBD429" name="X85FAD5E086DBD429"></a></p>
<h5>1.3-3 ConcentricFiltration</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConcentricFiltration</code>( <var class="Arg">K</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical complex <span class="SimpleMath">\(K\)</span> and integer <span class="SimpleMath">\(n \ge 1\)</span>, and returns a filtered pure cubical complex of filtration length <span class="SimpleMath">\(n\)</span>. The <span class="SimpleMath">\(t\)</span>-th term of the filtration is the intersection of <span class="SimpleMath">\(K\)</span> with the ball of radius <span class="SimpleMath">\(r_t\)</span> centred on the centre of gravity of <span class="SimpleMath">\(K\)</span>, where <span class="SimpleMath">\(0=r_1 \le r_2 \le r_3 \le \cdots \le r_n\)</span> are equally spaced rational numbers. The complex <span class="SimpleMath">\(K\)</span> is contained in the ball of radius <span class="SimpleMath">\(r_n\)</span>. (At present, this is implemented only for <span class="SimpleMath">\(2\)</span>- and <span class="SimpleMath">\(3\)</span>-dimensional complexes.)</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X861BA02C7902A4F4" name="X861BA02C7902A4F4"></a></p>
<h5>1.3-4 DirectProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectProduct</code>( <var class="Arg">M</var>, <var class="Arg">N</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">‣ DirectProduct</code>( <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two or more regular CW-complexes or two or more pure cubical complexes and returns their direct product.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap3.html">2</a></span> , <span class="URL"><a href="../tutorial/chap10.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCubical.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">7</a></span> </p>
<p><a id="X7DB4D3B57E0DA723" name="X7DB4D3B57E0DA723"></a></p>
<h5>1.3-5 FiltrationTerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FiltrationTerm</code>( <var class="Arg">K</var>, <var class="Arg">t</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">‣ FiltrationTerm</code>( <var class="Arg">K</var>, <var class="Arg">t</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a filtered regular CW-complex or a filtered pure cubical complex <span class="SimpleMath">\(K\)</span> together with an integer <span class="SimpleMath">\(t \ge 1\)</span>. The <span class="SimpleMath">\(t\)</span>-th term of the filtration is returned.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> </p>
<p><a id="X7B335342839E5146" name="X7B335342839E5146"></a></p>
<h5>1.3-6 Graph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Graph</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">‣ Graph</code>( <var class="Arg">K</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a regular CW-complex or a simplicial complex <span class="SimpleMath">\(K\)</span> and returns its <span class="SimpleMath">\(1\)</span>-skeleton as a graph.</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="../tutorial/chap7.html">4</a></span> , <span class="URL"><a href="../tutorial/chap10.html">5</a></span> , <span class="URL"><a href="../tutorial/chap11.html">6</a></span> , <span class="URL"><a href="../tutorial/chap14.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRandomComplexes.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">11</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGraphsOfGroups.html">12</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">13</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTopology.html">14</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">15</a></span> </p>
<p><a id="X7966519E78BC6C18" name="X7966519E78BC6C18"></a></p>
<h5>1.3-7 HomotopyGraph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomotopyGraph</code>( <var class="Arg">Y</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a regular CW-complex <span class="SimpleMath">\(Y\)</span> and returns a subgraph <span class="SimpleMath">\(M \subset Y^1\)</span> of the <span class="SimpleMath">\(1\)</span>-skeleton for which the induced homology homomorphisms <span class="SimpleMath">\(H_1(M,\mathbb Z) \rightarrow H_1(Y,\mathbb Z)\)</span> and <span class="SimpleMath">\(H_1(Y^1,\mathbb Z) \rightarrow H_1(Y,\mathbb Z)\)</span> have identical images. The construction tries to include as few edges in <span class="SimpleMath">\(M\)</span> as possible, though a minimum is not guaranteed.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> </p>
<p><a id="X84560FF678621AE1" name="X84560FF678621AE1"></a></p>
<h5>1.3-8 Nerve</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Nerve</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">‣ Nerve</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">‣ Nerve</code>( <var class="Arg">M</var>, <var class="Arg">n</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">‣ Nerve</code>( <var class="Arg">M</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</t | | |
