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1 Basic functionality for cellular complexes, fundamental groups and homology

This page covers the functions used in chapters 1 and 2 of the book An Invitation to Computational Homotopy.

1.1 Data ⟶ Cellular Complexes

1.1-1 RegularCWPolytope

Inputs a list L of vectors in R^n and outputs their convex hull as a regular CW-complex.

Inputs a permutation group G of degree d and vector v∈ R^d, and outputs the convex hull of the orbit {v^g : g∈ G} as a regular CW-complex.

Examples:

1.1-2 CubicalComplex

Inputs a binary array A and returns the cubical complex represented by A. The array A must of course be such that it represents a cubical complex.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

1.1-3 PureCubicalComplex

Inputs a binary array A and returns the pure cubical complex represented by A.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

1.1-4 PureCubicalKnot

Inputs integers n, k and returns the k-th prime knot on n crossings as a pure cubical complex (if this prime knot exists).

Inputs a list L describing an arc presentation for a knot or link and returns the knot or link as a pure cubical complex.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

1.1-5 PurePermutahedralKnot

Inputs integers n, k and returns the k-th prime knot on n crossings as a pure permutahedral complex (if this prime knot exists).

Inputs a list L describing an arc presentation for a knot or link and returns the knot or link as a pure permutahedral complex.

Examples: 1 , 2 

1.1-6 PurePermutahedralComplex

Inputs a binary array A and returns the pure permutahedral complex represented by A.

Examples: 1 , 2 , 3 , 4 

1.1-7 CayleyGraphOfGroup

Inputs a finite group G and a list L of elements in G.It returns the Cayley graph of the group generated by L.

Examples:

1.1-8 EquivariantEuclideanSpace

Inputs a crystallographic group G with left action on R^n together with a row vector v ∈ R^n. It returns an equivariant regular CW-space corresponding to the Dirichlet-Voronoi tessellation of R^n produced from the orbit of v under the action.

Examples: 1 

1.1-9 EquivariantOrbitPolytope

Inputs a permutation group G of degree n together with a row vector v ∈ R^n. It returns, as an equivariant regular CW-space, the convex hull of the orbit of v under the canonical left action of G on R^n.

Examples:

1.1-10 EquivariantTwoComplex

Inputs a suitable group G and returns, as an equivariant regular CW-space, the 2-complex associated to some presentation of G.

Examples: 1 

1.1-11 QuillenComplex

Inputs a finite group G and prime p, and returns the simplicial complex arising as the order complex of the poset of elementary abelian p-subgroups of G.

Examples: 1 , 2 , 3 , 4 

1.1-12 RestrictedEquivariantCWComplex

Inputs a G-equivariant regular CW-space Y and a subgroup H ≤ G for which GAP can find a transversal. It returns the equivariant regular CW-complex obtained by retricting the action to H.

Examples:

1.1-13 RandomSimplicialGraph

Inputs an integer n ≥ 1 and positive prime p, and returns an Erdős–Rényi random graph as a 1-dimensional simplicial complex. The graph has n vertices. Each pair of vertices is, with probability p, directly connected by an edge.

Examples: 1 

1.1-14 RandomSimplicialTwoComplex

Inputs an integer n ≥ 1 and positive prime p, and returns a Linial-Meshulam random simplicial 2-complex. The 1-skeleton of this simplicial complex is the complete graph on n vertices. Each triple of vertices lies, with probability p, in a common 2-simplex of the complex.

Examples: 1 , 2 

1.1-15 ReadCSVfileAsPureCubicalKnot

Reads a CSV file identified by a string str such as "file.pdb" or "path/file.pdb" and returns a 3-dimensional pure cubical complex K. Each line of the file should contain the coordinates of a point in R^3 and the complex K 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 K has the homotopy type of a circle.

If the test fails then try the function again with an integer r ≥ 2 entered as the optional second argument. The integer determines the resolution with which the knot is constructed.

The function can also read in a list L of strings identifying CSV files for several knots. In this case a list R of integer resolutions can also be entered. The lists L and R must be of equal length.

Examples: 1 

1.1-16 ReadImageAsPureCubicalComplex

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 t between 0 and 765. It returns a 2-dimensional pure cubical complex corresponding to a black/white version of the image determined by the threshold t. The 2-cells of the pure cubical complex correspond to pixels with RGB value R+G+B ≤ t.

Examples: 1 , 2 , 3 , 4 , 5 

1.1-17 ReadImageAsFilteredPureCubicalComplex

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 n. It returns a 2-dimensional filtered pure cubical complex of filtration length n. The kth term in the filtration is a pure cubical complex corresponding to a black/white version of the image determined by the threshold t_k=k × 765/n. The 2-cells of the kth term correspond to pixels with RGB value R+G+B ≤ t_k.

Examples: 1 

1.1-18 ReadImageAsWeightFunction

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 t. It constructs a 2-dimensional regular CW-complex Y from the image, together with a weight function w: Y→ Z corresponding to a filtration on Y of filtration length t. The pair [Y,w] is returned.

Examples:

1.1-19 ReadPDBfileAsPureCubicalComplex

Reads a PDB (Protein Database) file identified by a string str such as "file.pdb" or "path/file.pdb" and returns a 3-dimensional pure cubical complex K. The complex K should represent a (protein backbone) knot but this is not guaranteed. A useful check in this direction is to test that K has the homotopy type of a circle.

If the test fails then try the function again with an integer r ≥ 2 entered as the optional second argument. The integer determines the resolution with which the knot is constructed.

Examples: 1 , 2 , 3 

1.1-20 ReadPDBfileAsPurepermutahedralComplex

Reads a PDB (Protein Database) file identified by a string str such as "file.pdb" or "path/file.pdb" and returns a 3-dimensional pure permutahedral complex K. The complex K should represent a (protein backbone) knot but this is not guaranteed. A useful check in this direction is to test that K has the homotopy type of a circle.

If the test fails then try the function again with an integer r ≥ 2 entered as the optional second argument. The integer determines the resolution with which the knot is constructed.

Examples:

1.1-21 RegularCWPolytope

Inputs a list L of vectors in R^n and outputs their convex hull as a regular CW-complex.

Inputs a permutation group G of degree d and vector v∈ R^d, and outputs the convex hull of the orbit {v^g : g∈ G} as a regular CW-complex.

Examples:

1.1-22 SimplicialComplex

Inputs a list L 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 L can also contain non-maximal simplices.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

1.1-23 SymmetricMatrixToFilteredGraph

Inputs an n × n symmetric matrix A, a positive integer m and a positive rational s. The function returns a filtered graph of filtration length m. The t-th term of the filtration is a graph with n vertices and an edge between the i-th and j-th vertices if the (i,j) entry of A is less than or equal to t × s/m.

If the optional input s is omitted then it is set equal to the largest entry in the matrix A.

Examples: 1 , 2 , 3 

1.1-24 SymmetricMatrixToGraph

Inputs an n× n symmetric matrix A over the rationals and a rational number t ≥ 0, and returns the graph on the vertices 1,2, ..., n with an edge between distinct vertices i and j precisely when the (i,j) entry of A is ≤ t.

Examples: 1 , 2 

1.2 Metric Spaces

1.2-1 CayleyMetric

Inputs two permutations g,h and optionally the degree N of a symmetric group containing them. It returns the minimum number of transpositions needed to express g*h^-1 as a product of transpositions.

Examples: 1 

1.2-2 EuclideanMetric

Inputs two vectors v,w ∈ R^n and returns a rational number approximating the Euclidean distance between them.

Examples:

1.2-3 EuclideanSquaredMetric

Inputs two vectors v,w ∈ R^n and returns the square of the Euclidean distance between them.

Examples:

1.2-4 HammingMetric

Inputs two permutations g,h and optionally the degree N of a symmetric group containing them. It returns the minimum number of integers moved by the permutation g*h^-1.

Examples:

1.2-5 KendallMetric

Inputs two permutations g,h and optionally the degree N of a symmetric group containing them. It returns the minimum number of adjacent transpositions needed to express g*h^-1 as a product of adjacent transpositions. An adjacent transposition is of the form (i,i+1).

Examples:

1.2-6 ManhattanMetric

Inputs two vectors v,w ∈ R^n and returns the Manhattan distance between them.

Examples: 1 

1.2-7 VectorsToSymmetricMatrix

Inputs a list V ={ v_1, ..., v_k} ∈ R^n and returns the k × k symmetric matrix of Euclidean distances d(v_i, v_j). When these distances are irrational they are approximated by a rational number.

As an optional second argument any rational valued function d(x,y) can be entered.

Examples: 1 , 2 , 3 

1.3 Cellular Complexes ⟶ Cellular Complexes

1.3-1 BoundaryMap

Inputs a pure regular CW-complex K and returns the regular CW-inclusion map ι : ∂ K ↪ K from the boundary ∂ K into the complex K.

Examples: 1 , 2 , 3 

1.3-2 CliqueComplex

Inputs a graph G and integer n ≥ 1. It returns the n-skeleton of a simplicial complex K with one k-simplex for each complete subgraph of G on k+1 vertices.

Inputs a fitered graph F and integer n ≥ 1. It returns the n-skeleton of a filtered simplicial complex K whose t-term has one k-simplex for each complete subgraph of the t-th term of G on k+1 vertices.

Inputs a simplicial complex of dimension d=1 or d=2. If d=1 then the clique complex of a graph returned. If d=2 then the clique complex of a 2-complex is returned.

Examples: 1 

1.3-3 ConcentricFiltration

Inputs a pure cubical complex K and integer n ≥ 1, and returns a filtered pure cubical complex of filtration length n. The t-th term of the filtration is the intersection of K with the ball of radius r_t centred on the centre of gravity of K, where 0=r_1 ≤ r_2 ≤ r_3 ≤ ⋯ ≤ r_n are equally spaced rational numbers. The complex K is contained in the ball of radius r_n. (At present, this is implemented only for 2- and 3-dimensional complexes.)

Examples:

1.3-4 DirectProduct

Inputs two or more regular CW-complexes or two or more pure cubical complexes and returns their direct product.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

1.3-5 FiltrationTerm

Inputs a filtered regular CW-complex or a filtered pure cubical complex K together with an integer t ≥ 1. The t-th term of the filtration is returned.

Examples: 1 

1.3-6 Graph

Inputs a regular CW-complex or a simplicial complex K and returns its 1-skeleton as a graph.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 

1.3-7 HomotopyGraph

Inputs a regular CW-complex Y and returns a subgraph M ⊂ Y^1 of the 1-skeleton for which the induced homology homomorphisms H_1(M, Z) → H_1(Y, Z) and H_1(Y^1, Z) → H_1(Y, Z) have identical images. The construction tries to include as few edges in M as possible, though a minimum is not guaranteed.

Examples: 1 

1.3-8 Nerve

Inputs a pure cubical complex or pure permutahedral complex M and returns the simplicial complex K obtained by taking the nerve of an open cover of |M|, the open sets in the cover being sufficiently small neighbourhoods of the top-dimensional cells of |M|. The spaces |M| and |K| are homotopy equivalent by the Nerve Theorem. If an integer n ≥ 0 is supplied as the second argument then only the n-skeleton of K is returned.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 

1.3-9 RegularCWComplex

Inputs a simplicial, pure cubical, cubical or pure permutahedral complex K and returns the corresponding regular CW-complex. Inputs a list L=Y!.boundaries of boundary incidences of a regular CW-complex Y and returns Y. Inputs a list L=Y!.boundaries of boundary incidences of a regular CW-complex Y together with a list M=Y!.orientation of incidence numbers and returns a regular CW-complex Y. The availability of precomputed incidence numbers saves recalculating them.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

1.3-10 RegularCWMap

Inputs a pure cubical complex M and a subcomplex A and returns the inclusion map A → M as a map of regular CW complexes.

Examples: 1 , 2 , 3 

1.3-11 ThickeningFiltration

Inputs a pure cubical complex K and integer n ≥ 1, and returns a filtered pure cubical complex of filtration length n. The t-th term of the filtration is the t-fold thickening of K. If an integer s ≥ 1 is entered as the optional third argument then the t-th term of the filtration is the ts-fold thickening of K.

Examples: 1 , 2 

1.4 Cellular Complexes ⟶ Cellular Complexes (Preserving Data Types)

1.4-1 ContractedComplex

Inputs a complex (regular CW, Filtered regular CW, pure cubical etc.) and returns a homotopy equivalent subcomplex.

Inputs a pure cubical complex or pure permutahedral complex K and a subcomplex S. It returns a homotopy equivalent subcomplex of K that contains S.

Inputs a graph G and returns a subgraph S such that the clique complexes of G and S are homotopy equivalent.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

1.4-2 ContractibleSubcomplex

Inputs a non-empty pure cubical, pure permutahedral or simplicial complex K and returns a contractible subcomplex.

Examples: 1 , 2 

1.4-3 KnotReflection

Inputs a pure cubical knot and returns the reflected knot.

Examples:

1.4-4 KnotSum

Inputs two pure cubical knots and returns their sum.

Examples: 1 , 2 , 3 , 4 , 5 

1.4-5 OrientRegularCWComplex

Inputs a regular CW-complex Y and computes and stores incidence numbers for Y. If Y already has incidence numbers then the function does nothing.

Examples:

1.4-6 PathComponent

Inputs a simplicial, pure cubical or pure permutahedral complex K together with an integer 1 ≤ n ≤ β_0(K). The n-th path component of K is returned.

Examples: 1 , 2 , 3 

1.4-7 PureComplexBoundary

Inputs a d-dimensional pure cubical or pure permutahedral complex M and returns a d-dimensional complex consisting of the closure of those d-cells whose boundaries contains some cell with coboundary of size less than the maximal possible size.

Examples: 1 

1.4-8 PureComplexComplement

Inputs a pure cubical complex or a pure permutahedral complex and returns its complement.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

1.4-9 PureComplexDifference

Inputs two pure cubical complexes or two pure permutahedral complexes and returns the difference M - N.

Examples: 1 

1.4-10 PureComplexInterstection

Inputs two pure cubical complexes or two pure permutahedral complexes and returns their intersection.

Examples:

1.4-11 PureComplexThickened

Inputs a pure cubical complex or a pure permutahedral complex and returns the a thickened complex.

Examples: 1 

1.4-12 PureComplexUnion

Inputs two pure cubical complexes or two pure permutahedral complexes and returns their union.

Examples: 1 

1.4-13 SimplifiedComplex

Inputs a regular CW-complex or a pure permutahedral complex K and returns a homeomorphic complex with possibly fewer cells and certainly no more cells.

Inputs a free ZG-resolution R of Z and returns a ZG-resolution S with potentially fewer free generators.

Inputs a chain complex C of free abelian groups and returns a chain homotopic chain complex D with potentially fewer free generators.

Examples: 1 , 2 , 3 , 4 , 5 , 6 

1.4-14 ZigZagContractedComplex

Inputs a pure cubical, filtered pure cubical or pure permutahedral complex and returns a homotopy equivalent complex. In the filtered case, the t-th term of the output is homotopy equivalent to the t-th term of the input for all t.

Examples: 1 

1.5 Cellular Complexes ⟶ Homotopy Invariants

1.5-1 AlexanderPolynomial

Inputs a 3-dimensional pure cubical or pure permutahdral complex K representing a knot and returns the Alexander polynomial of the fundamental group G = π_1( R^3∖ K).

Inputs a finitely presented group G with infinite cyclic abelianization and returns its Alexander polynomial.

Examples: 1 , 2 , 3 , 4 

1.5-2 BettiNumber

Inputs a simplicial, cubical, pure cubical, pure permutahedral, regular CW, chain or sparse chain complex K together with an integer n ≥ 0 and returns the nth Betti number of K.

Inputs a simplicial, cubical, pure cubical, pure permutahedral or regular CW-complex K together with an integer n ≥ 0 and a prime p ≥ 0 or p=0. In this case the nth Betti number of K over a field of characteristic p is returned.

Examples: 1 

1.5-3 EulerCharacteristic

Inputs a chain complex C and returns its Euler characteristic.

Inputs a cubical, or pure cubical, or pure permutahedral or regular CW-, or simplicial complex K and returns its Euler characteristic.

Examples:

1.5-4 EulerIntegral

Inputs a regular CW-complex Y and a weight function w: Y→ Z, and returns the Euler integral ∫_Y w dχ.

Examples:

1.5-5 FundamentalGroup

Inputs a regular CW, simplicial, pure cubical or pure permutahedral complex K and returns the fundamental group.

Inputs a regular CW complex K and the number n of some zero cell. It returns the fundamental group of K based at the n-th zero cell.

Inputs a regular CW map F and returns the induced homomorphism of fundamental groups. If the number of some zero cell in the domain of F is entered as an optional second variable then the fundamental group is based at this zero cell.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 

1.5-6 FundamentalGroupOfQuotient

Inputs a G-equivariant regular CW complex Y and returns the group G.

Examples: 1 

1.5-7 IsAspherical

Inputs a free group F and a list R of words in F. The function attempts to test if the quotient group G=F/⟨ R ⟩^F is aspherical. If it succeeds it returns true. Otherwise the test is inconclusive and fail is returned.

Examples: 1 , 2 , 3 , 4 

1.5-8 KnotGroup

Inputs a pure cubical or pure permutahedral complex K and returns the fundamental group of its complement. If the complement is path-connected then this fundamental group is unique up to isomorphism. Otherwise it will depend on the path-component in which the randomly chosen base-point lies.

Examples: 1 

1.5-9 PiZero

Inputs a regular CW-complex Y, or graph Y, or simplicial complex Y and returns a pair [cells,r] where: cells is a list of vertices of Y representing the distinct path-components; r(v) is a function which, for each vertex v of Y returns the representative vertex r(v) ∈ cells.

Examples: 1 

1.5-10 PersistentBettiNumbers

Inputs a filtered simplicial, filtered pure cubical, filtered regular CW, filtered chain or filtered sparse chain complex K together with an integer n ≥ 0 and returns the nth PersistentBetti numbers of K as a list of lists of integers.

Inputs a filtered simplicial, filtered pure cubical, filtered regular CW, filtered chain or filtered sparse chain complex K together with an integer n ≥ 0 and a prime p ≥ 0 or p=0. In this case the nth PersistentBetti numbers of K over a field of characteristic p are returned.

Examples: 1 

1.6 Data ⟶ Homotopy Invariants

1.6-1 DendrogramMat

Inputs an n× n symmetric matrix A over the rationals, a rational t ≥ 0 and an integer s ≥ 1. A list [v_1, ..., v_t+1] is returned with each v_k a list of positive integers. Let t_k = (k-1)s. Let G(A,t_k) denote the graph with vertices 1, ..., n and with distinct vertices i and j connected by an edge when the (i,j) entry of A is ≤ t_k. The i-th path component of G(A,t_k) is included in the v_k[i]-th path component of G(A,t_k+1). This defines the integer vector v_k. The vector v_k has length equal to the number of path components of G(A,t_k).

Examples:

1.7 Cellular Complexes ⟶ Non Homotopy Invariants

1.7-1 ChainComplex

Inputs a cubical, or pure cubical, or pure permutahedral or simplicial complex K and returns its chain complex of free abelian groups. In degree n this chain complex has one free generator for each n-dimensional cell of K.

Inputs a regular CW-complex Y and returns a chain complex C which is chain homotopy equivalent to the cellular chain complex of Y. In degree n the free abelian chain group C_n has one free generator for each critical n-dimensional cell of Y with respect to some discrete vector field on Y.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 

1.7-2 ChainComplexEquivalence

Inputs a regular CW-complex X and returns a pair [f_∗, g_∗] of chain maps f_∗: C_∗(X) → D_∗(X), g_∗: D_∗(X) → C_∗(X). Here C_∗(X) is the standard cellular chain complex of X with one free generator for each cell in X. The chain complex D_∗(X) is a typically smaller chain complex arising from a discrete vector field on X. The chain maps f_∗, g_∗ are chain homotopy equivalences.

Examples:

1.7-3 ChainComplexOfQuotient

Inputs a G-equivariant regular CW-complex Y and returns the cellular chain complex of the quotient space Y/G.

Examples: 1 

1.7-4 ChainMap

Inputs a pure cubical complex Y and pure cubical sucomplexes X⊂ Y, B⊂ Y,A⊂ B. It returns the induced chain map f_∗: C_∗(X/A) → C_∗(Y/B) of cellular chain complexes of pairs. (Typlically one takes A and B to be empty or contractible subspaces, in which case C_∗(X/A) ≃ C_∗(X), C_∗(Y/B) ≃ C_∗(Y).)

Inputs a map f: X → Y between two regular CW-complexes X,Y and returns an induced chain map f_∗: C_∗(X) → C_∗(Y) where C_∗(X), C_∗(Y) are chain homotopic to (but usually smaller than) the cellular chain complexes of X, Y.

Inputs a map f: X → Y between two simplicial complexes X,Y and returns the induced chain map f_∗: C_∗(X) → C_∗(Y) of cellular chain complexes.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

1.7-5 CochainComplex

Inputs a cubical, or pure cubical, or pure permutahedral or simplicial complex K and returns its cochain complex of free abelian groups. In degree n this cochain complex has one free generator for each n-dimensional cell of K.

Inputs a regular CW-complex Y and returns a cochain complex C which is chain homotopy equivalent to the cellular cochain complex of Y. In degree n the free abelian cochain group C_n has one free generator for each critical n-dimensional cell of Y with respect to some discrete vector field on Y.

Examples:

1.7-6 CriticalCells

Inputs a regular CW-complex K and returns its critical cells with respect to some discrete vector field on K. If no discrete vector field on K is available then one will be computed and stored.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

1.7-7 DiagonalApproximation

Inputs a regular CW-complex X and outputs a pair [p,ι] of maps of CW-complexes. The map p: X^∆ → X will often be a homotopy equivalence. This is always the case if X is the CW-space of any pure cubical complex. In general, one can test to see if the induced chain map p_∗ : C_∗(X^∆) → C_∗(X) is an isomorphism on integral homology. The second map ι : X^∆ ↪ X× X is an inclusion into the direct product. If p_∗ induces an isomorphism on homology then the chain map ι_∗: C_∗(X^∆) → C_∗(X× X) can be used to compute the cup product.

Examples: 1 

1.7-8 Size

Inputs a regular CW complex or a simplicial complex Y and returns the number of cells in the complex.

Inputs a d-dimensional pure cubical or pure permutahedral complex K and returns the number of d-dimensional cells in the complex.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 

1.8 (Co)chain Complexes ⟶ (Co)chain Complexes

1.8-1 FilteredTensorWithIntegers

Inputs a free ZG-resolution R for which "filteredDimension" lies in NamesOfComponents(R). (Such a resolution can be produced using TwisterTensorProduct(), ResolutionNormalSubgroups() or FreeGResolution().) It returns the filtered chain complex obtained by tensoring with the trivial module Z.

Examples: 1 , 2 

1.8-2 FilteredTensorWithIntegersModP

Inputs a free ZG-resolution R for which "filteredDimension" lies in NamesOfComponents(R), together with a prime p. (Such a resolution can be produced using TwisterTensorProduct(), ResolutionNormalSubgroups() or FreeGResolution().) It returns the filtered chain complex obtained by tensoring with the trivial module F, the field of p elements.

Examples: 1 , 2 

1.8-3 HomToIntegers

Inputs a chain complex C of free abelian groups and returns the cochain complex Hom_ Z(C, Z).

Inputs a free ZG-resolution R in characteristic 0 and returns the cochain complex Hom_ ZG(R, Z).

Inputs an equivariant chain map F: R→ S of resolutions and returns the induced cochain map Hom_ ZG(S, Z) ⟶ Hom_ ZG(R, Z).

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 

1.8-4 TensorWithIntegersModP

Inputs a chain complex C of characteristic 0 and a prime integer p. It returns the chain complex C ⊗_ Z Z_p of characteristic p.

Inputs a free ZG-resolution R of characteristic 0 and a prime integer p. It returns the chain complex R ⊗_ ZG Z_p of characteristic p.

Inputs an equivariant chain map F: R → S in characteristic 0 a prime integer p. It returns the induced chain map F⊗_ ZG Z_p : R ⊗_ ZG Z_p ⟶ S ⊗_ ZG Z_p.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 

1.9 (Co)chain Complexes ⟶ Homotopy Invariants

1.9-1 Cohomology

Inputs a cochain complex C and integer n ≥ 0 and returns the n-th cohomology group of C as a list of its abelian invariants.

Inputs a chain map F and integer n ≥ 0. It returns the induced cohomology homomorphism H_n(F) as a homomorphism of finitely presented groups.

Inputs a cubical, or pure cubical, or pure permutahedral or regular CW or simplicial complex K together with an integer n ≥ 0. It returns the n-th integral cohomology group of K as a list of its abelian invariants.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 

1.9-2 CupProduct

Inputs a regular CW-complex Y and returns a function f(p,q,P,Q). This function f inputs two integers p,q ≥ 0 and two integer lists P=[p_1, ..., p_m], Q=[q_1, ..., q_n] representing elements P∈ H^p(Y, Z) and Q∈ H^q(Y, Z). The function f returns a list P ∪ Q representing the cup product P ∪ Q ∈ H^p+q(Y, Z).

Inputs a free ZG resolution R of Z for some group G, together with integers p,q ≥ 0 and integer lists P, Q representing cohomology classes P∈ H^p(G, Z), Q∈ H^q(G, Z). An integer list representing the cup product P∪ Q ∈ H^p+q(G, Z) is returned.

Examples: 1 , 2 , 3 , 4 , 5 

1.9-3 Homology

Inputs a chain complex C and integer n ≥ 0 and returns the n-th homology group of C as a list of its abelian invariants.

Inputs a chain map F and integer n ≥ 0. It returns the induced homology homomorphism H_n(F) as a homomorphism of finitely presented groups.

Inputs a cubical, or pure cubical, or pure permutahedral or regular CW or simplicial complex K together with an integer n ≥ 0. It returns the n-th integral homology group of K as a list of its abelian invariants.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

1.10 Visualization

1.10-1 BarCodeDisplay

Displays a barcode L=PersitentBettiNumbers(X,n).

Examples: 1 , 2 

1.10-2 BarCodeCompactDisplay

Displays a barcode L=PersitentBettiNumbers(X,n) in compact form.

Examples: 1 , 2 , 3 

1.10-3 CayleyGraphOfGroup

Inputs a finite group G and a list L of elements in G.It displays the Cayley graph of the group generated by L where edge colours correspond to generators.

Examples:

1.10-4 Display

Displays a graph G; a 2- or 3-dimensional pure cubical complex M; a 3-dimensional pure permutahedral complex M.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 

1.10-5 DisplayArcPresentation

Displays a 3-dimensional pure cubical knot K=PureCubicalKnot(L) in the form of an arc presentation.

Examples:

1.10-6 DisplayCSVKnotFile

Inputs a string str that identifies a csv file containing the points on a piecewise linear knot in R^3. It displays the knot.

Examples:

1.10-7 DisplayDendrogram

Displays the dendrogram L:=DendrogramMat(A,t,s).

Examples:

1.10-8 DisplayDendrogramMat

Inputs an n× n symmetric matrix A over the rationals, a rational t ≥ 0 and an integer s ≥ 1. The dendrogram defined by DendrogramMat(A,t,s) is displayed.

Examples:

1.10-9 DisplayPDBfile

Displays the protein backone described in a PDB (Protein Database) file identified by a string str such as "file.pdb" or "path/file.pdb".

Examples: 1 

1.10-10 OrbitPolytope

Inputs a permutation group or finite matrix group G of degree d and a rational vector v∈ R^d. In both cases there is a natural action of G on R^d. Let P(G,v) be the convex hull of the orbit of v under the action of G. The function also inputs a sublist L of the following list of strings: ["dimension","vertex_degree", "visual_graph", "schlegel", "visual"]

Depending on L, the function displays the following information:
the dimension of the orbit polytope P(G,v);
the degree of a vertex in the graph of P(G,v);
a visualization of the graph of P(G,v);
a visualization of the Schlegel diagram of P(G,v);
a visualization of the polytope P(G,v) if d=2,3.

The function requires Polymake software.

Examples: 1 , 2 

1.10-11 ScatterPlot

Inputs a list L=[[x_1,y_1],..., [x_n,y_n]] of pairs of rational numbers and displays a scatter plot of the points in the x-y-plane.

Examples: 1 

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