<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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 <spanclass="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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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 <spanclass="SimpleMath">\((i,i+1)\)</span>.</p>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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 <spanclass="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>
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
