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1 Cellular complexes

### 1 Cellular complexes

Data ⟶ Cellular Complexes

    RegularCWPolytope(L):: List --> RegCWComplex

    RegularCWPolytope(G,v):: PermGroup, List --> RegCWComplex

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.

    CubicalComplex(A):: List --> 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.

    PureCubicalComplex(A):: List --> PureCubicalComplex

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

    PureCubicalKnot(n,k):: Int, Int --> PureCubicalComplex

    PureCubicalKnot(L):: List --> PureCubicalComplex

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.

    PurePermutahedralKnot(n,k):: Int, Int --> PurePermutahedralComplex

    PurePermutahedralKnot(L):: List --> PurePermutahedralComplex

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.

    PurePermutahedralComplex(A):: List --> PurePermComplex

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

    CayleyGraphOfGroup(G,L):: Group, List --> Graph

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

    EquivariantEuclideanSpace(G,v):: MatrixGroup, List --> EquivariantRegCWComplex

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.

    EquivariantOrbitPolytope(G,v):: PermGroup, List --> EquivariantRegCWComplex

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.

    EquivariantTwoComplex(G):: Group --> EquivariantRegCWComplex

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

    QuillenComplex(G,p):: Group, Int --> SimplicialComplex

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.

    RestrictedEquivariantCWComplex(Y,H):: RegCWComplex, Group --> EquivariantRegCWComplex

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

    RandomSimplicialGraph(n,p):: Int, Int --> SimplicialComplex

Inputs an integer n ge 1 and positive prime p, and returns an Erd\"os-R\'enyi 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.

    RandomSimplicialTwoComplex(n,p):: Int, Int --> SimplicialComplex

Inputs an integer n ge 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.

    ReadCSVfileAsPureCubicalKnot(str):: String --> PureCubicalComplex

    ReadCSVfileAsPureCubicalKnot(str,r):: String, Int --> PureCubicalComplex

    ReadCSVfileAsPureCubicalKnot(L):: List --> PureCubicalComplex

    ReadCSVfileAsPureCubicalKnot(L,R):: List,List --> PureCubicalComplex

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 ge 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.

    ReadImageAsPureCubicalComplex(str,t):: String, Int --> PureCubicalComplex

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 le t.

    ReadImageAsFilteredPureCubicalComplex(str,n):: String, Int --> FilteredPureCubicalComplex

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 le t_k.

    ReadImageAsWeightFunction(str,t):: String, Int --> RegCWComplex, Function

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.

    ReadPDBfileAsPureCubicalComplex(str):: String --> PureCubicalComplex

    ReadPDBfileAsPureCubicalComplex(str,r):: String, Int --> PureCubicalComplex

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 ge 2 entered as the optional second argument. The integer determines the resolution with which the knot is constructed.

    ReadPDBfileAsPurepermutahedralComplex(str):: String --> PurePermComplex

    ReadPDBfileAsPurePermutahedralComplex(str,r):: String, Int --> PurePermComplex

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 ge 2 entered as the optional second argument. The integer determines the resolution with which the knot is constructed.

    RegularCWPolytope(L):: List --> RegCWComplex

    RegularCWPolytope(G,v):: PermGroup, List --> RegCWComplex

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.

    SimplicialComplex(L):: List --> 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.

    SymmetricMatrixToFilteredGraph(A,m,s):: Mat, Int, Rat --> FilteredGraph

    SymmetricMatrixToFilteredGraph(A,m):: Mat, Int --> FilteredGraph

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.

    SymmetricMatrixToGraph(A,t):: Mat, Rat --> Graph

Inputs an n× n symmetric matrix A over the rationals and a rational number t ge 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 le t.

Metric Spaces

    CayleyMetric(g,h):: Permutation, Permutation --> Int

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.

    EuclideanMetric(g,h):: List, List --> Rat

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

    EuclideanSquaredMetric(g,h):: List, List --> Rat

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

    HammingMetric(g,h):: Permutation, Permutation --> Int

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.

    KendallMetric(g,h):: Permutation, Permutation --> Int

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 {\em adjacent} transposition is of the form (i,i+1).

    ManhattanMetric(g,h):: List, List --> Rat

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

    VectorsToSymmetricMatrix(V):: List --> Matrix

    VectorsToSymmetricMatrix(V,d):: List, Function --> Matrix

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.

Cellular Complexes ⟶ Cellular Complexes

    BoundaryMap(K):: RegCWComplex --> RegCWMap

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.

    CliqueComplex(G,n):: Graph, Int --> SimplicialComplex

    CliqueComplex(F,n):: FilteredGraph, Int --> FilteredSimplicialComplex

    CliqueComplex(K,n):: SimplicialComplex, Int --> SimplicialComplex

Inputs a graph G and integer n ge 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 ge 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.

    ConcentricFiltration(K,n):: PureCubicalComplex, Int --> FilteredPureCubicalComplex

Inputs a pure cubical complex K and integer n ge 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 le r_2 le r_3 le ⋯ le 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.)

    DirectProduct(M,N):: RegCWComplex, RegCWComplex --> RegCWComplex

    DirectProduct(M,N):: PureCubicalComplex, PureCubicalComplex --> PureCubicalComplex

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

    FiltrationTerm(K,t):: FilteredPureCubicalComplex, Int --> PureCubicalComplex

    FiltrationTerm(K,t):: FilteredRegCWComplex, Int --> RegCWComplex

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

    Graph(K):: RegCWComplex --> Graph

    Graph(K):: SimplicialComplex --> Graph

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

    HomotopyGraph(Y):: RegCWComplex --> Graph

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.

    Nerve(M):: PureCubicalComplex --> SimplicialComplex

    Nerve(M):: PurePermComplex --> SimplicialComplex

    Nerve(M,n):: PureCubicalComplex, Int --> SimplicialComplex

    Nerve(M,n):: PurePermComplex, Int --> SimplicialComplex

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 ge 0 is supplied as the second argument then only the n-skeleton of K is returned.

    RegularCWComplex(K):: SimplicialComplex --> RegCWComplex

    RegularCWComplex(K):: PureCubicalComplex --> RegCWComplex

    RegularCWComplex(K):: CubicalComplex --> RegCWComplex

    RegularCWComplex(K):: PurePermComplex --> RegCWComplex

    RegularCWComplex(L):: List --> RegCWComplex

    RegularCWComplex(L,M):: List,List --> RegCWComplex

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.

    RegularCWMap(M,A):: PureCubicalComplex, PureCubicalComplex --> RegCWMap

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

    ThickeningFiltration(K,n):: PureCubicalComplex, Int --> FilteredPureCubicalComplex

    ThickeningFiltration(K,n,s):: PureCubicalComplex, Int, Int --> FilteredPureCubicalComplex

Inputs a pure cubical complex K and integer n ge 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 ge 1 is entered as the optional third argument then the t-th term of the filtration is the ts-fold thickening of K.

Cellular Complexes ⟶ Cellular Complexes (Preserving Data Types)

    ContractedComplex(K):: RegularCWComplex --> RegularCWComplex

    ContractedComplex(K):: FilteredRegularCWComplex --> FilteredRegularCWComplex

    ContractedComplex(K):: CubicalComplex --> CubicalComplex

    ContractedComplex(K):: PureCubicalComplex --> PureCubicalComplex

    ContractedComplex(K,S):: PureCubicalComplex, PureCubicalComplex --> PureCubicalComplex

    ContractedComplex(K):: FilteredPureCubicalComplex --> FilteredPureCubicalComplex

    ContractedComplex(K):: PurePermComplex --> PurePermComplex

    ContractedComplex(K,S):: PurePermComplex, PurePermComplex --> PurePermComplex

    ContractedComplex(K):: SimplicialComplex --> SimplicialComplex

    ContractedComplex(G):: Graph --> Graph

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.

    ContractibleSubcomplex(K):: PureCubicalComplex --> PureCubicalComplex

    ContractibleSubcomplex(K):: PurePermComplex --> PurePermComplex

    ContractibleSubcomplex(K):: SimplicialComplex --> SimplicialComplex

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

    KnotReflection(K):: PureCubicalComplex --> PureCubicalComplex

Inputs a pure cubical knot and returns the reflected knot.

    KnotSum(K,L):: PureCubicalComplex, PureCubicalComplex --> PureCubicalComplex

Inputs two pure cubical knots and returns their sum.

    OrientRegularCWComplex(Y):: RegCWComplex --> Void

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.

    PathComponent(K,n):: SimplicialComplex, Int --> SimplicialComplex

    PathComponent(K,n):: PureCubicalComplex, Int --> PureCubicalComplex

    PathComponent(K,n):: PurePermComplex, Int --> PurePermComplex

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

    PureComplexBoundary(M):: PureCubicalComplex --> PureCubicalComplex

    PureComplexBoundary(M):: PurePermComplex --> PurePermComplex

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.

    PureComplexComplement(M):: PureCubicalComplex --> PureCubicalComplex

    PureComplexComplement(M):: PurePermComplex --> PurePermComplex

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

    PureComplexDifference(M,N):: PureCubicalComplex, PureCubicalComplex --> PureCubicalComplex

    PureComplexDifference(M,N):: PurePermComplex, PurePermComplex --> PurePermComplex

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

    PureComplexInterstection(M,N):: PureCubicalComplex, PureCubicalComplex --> PureCubicalComplex

    PureComplexIntersection(M,N):: PurePermComplex, PurePermComplex --> PurePermComplex

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

    PureComplexThickened(M):: PureCubicalComplex --> PureCubicalComplex

    PureComplexThickened(M):: PurePermComplex --> PurePermComplex

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

    PureComplexUnion(M,N):: PureCubicalComplex, PureCubicalComplex --> PureCubicalComplex

    PureComplexUnion(M,N):: PurePermComplex, PurePermComplex --> PurePermComplex

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

    SimplifiedComplex(K):: RegularCWComplex --> RegularCWComplex

    SimplifiedComplex(K):: PurePermComplex --> PurePermComplex

    SimplifiedComplex(R):: FreeResolution --> FreeResolution

    SimplifiedComplex(C):: ChainComplex --> ChainComplex

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.

    ZigZagContractedComplex(K):: PureCubicalComplex --> PureCubicalComplex

    ZigZagContractedComplex(K):: FilteredPureCubicalComplex --> FilteredPureCubicalComplex

    ZigZagContractedComplex(K):: PurePermComplex --> PurePermComplex

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.

Cellular Complexes ⟶ Homotopy Invariants

    AlexanderPolynomial(K):: PureCubicalComplex --> Polynomial

    AlexanderPolynomial(K):: PurePermComplex --> Polynomial

    AlexanderPolynomial(G):: FpGroup --> Polynomial

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.

    BettiNumber(K,n):: SimplicialComplex, Int --> Int

    BettiNumber(K,n):: PureCubicalComplex, Int --> Int

    BettiNumber(K,n):: CubicalComplex, Int --> Int

    BettiNumber(K,n):: PurePermComplex, Int --> Int

    BettiNumber(K,n):: RegCWComplex, Int --> Int

    BettiNumber(K,n):: ChainComplex, Int --> Int

    BettiNumber(K,n):: SparseChainComplex, Int --> Int

    BettiNumber(K,n,p):: SimplicialComplex, Int, Int --> Int

    BettiNumber(K,n,p):: PureCubicalComplex, Int, Int --> Int

    BettiNumber(K,n,p):: CubicalComplex, Int, Int --> Int

    BettiNumber(K,n,p):: PurePermComplex, Int, Int --> Int

    BettiNumber(K,n,p):: RegCWComplex, Int, Int --> Int

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

    EulerCharacteristic(C):: ChainComplex --> Int

    EulerCharacteristic(K):: CubicalComplex --> Int

    EulerCharacteristic(K):: PureCubicalComplex --> Int

    EulerCharacteristic(K):: PurePermComplex --> Int

    EulerCharacteristic(K):: RegCWComplex --> Int

    EulerCharacteristic(K):: SimplicialComplex --> Int

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.

    EulerIntegral(Y,w):: RegCWComplex, Int --> Int

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

    FundamentalGroup(K):: RegCWComplex --> FpGroup

    FundamentalGroup(K,n):: RegCWComplex, Int --> FpGroup

    FundamentalGroup(K):: SimplicialComplex --> FpGroup

    FundamentalGroup(K):: PureCubicalComplex --> FpGroup

    FundamentalGroup(K):: PurePermComplex --> FpGroup

    FundamentalGroup(F):: RegCWMap --> GroupHomomorphism

    FundamentalGroup(F,n):: RegCWMap, Int --> GroupHomomorphism

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.

    FundamentalGroupOfQuotient(Y):: EquivariantRegCWComplex --> Group

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

    IsAspherical(F,R):: FreeGroup, List --> Boolean

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.

    KnotGroup(K):: PureCubicalComplex --> FpGroup

    KnotGroup(K):: PureCubicalComplex --> FpGroup

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.

    PiZero(Y):: RegCWComplex --> List

    PiZero(Y):: Graph --> List

    PiZero(Y):: SimplicialComplex --> List

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.

    PersistentBettiNumbers(K,n):: FilteredSimplicialComplex, Int --> List

    PersistentBettiNumbers(K,n):: FilteredPureCubicalComplex, Int --> List

    PersistentBettiNumbers(K,n):: FilteredRegCWComplex, Int --> List

    PersistentBettiNumbers(K,n):: FilteredChainComplex, Int --> List

    PersistentBettiNumbers(K,n):: FilteredSparseChainComplex, Int --> List

    PersistentBettiNumbers(K,n,p):: FilteredSimplicialComplex, Int, Int --> List

    PersistentBettiNumbers(K,n,p):: FilteredPureCubicalComplex, Int, Int --> List

    PersistentBettiNumbers(K,n,p):: FilteredRegCWComplex, Int, Int --> List

    PersistentBettiNumbers(K,n,p):: FilteredChainComplex, Int, Int --> List

    PersistentBettiNumbers(K,n,p):: FilteredSparseChainComplex, Int, Int --> List

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

Data ⟶ Homotopy Invariants

    DendrogramMat(A,t,s):: Mat, Rat, Int --> List

Inputs an n× n symmetric matrix A over the rationals, a rational t ge 0 and an integer s ge 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 le 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).

Cellular Complexes ⟶ Non Homotopy Invariants

    ChainComplex(K):: CubicalComplex --> ChainComplex

    ChainComplex(K):: PureCubicalComplex --> ChainComplex

    ChainComplex(K):: PurePermComplex --> ChainComplex

    ChainComplex(Y):: RegCWComplex --> ChainComplex

    ChainComplex(K):: SimplicialComplex --> 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.

    ChainComplexEquivalence(X):: RegCWComplex --> List

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.

    ChainComplexOfQuotient(Y):: EquivariantRegCWComplex --> ChainComplex

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

    ChainMap(X,A,Y,B):: PureCubicalComplex, PureCubicalComplex, PureCubicalComplex, PureCubicalComplex --> ChainMap

    ChainMap(f):: RegCWMap --> ChainMap

    ChainMap(f):: SimplicialMap --> ChainComplex

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.

    CochainComplex(K):: CubicalComplex --> CochainComplex

    CochainComplex(K):: PureCubicalComplex --> CochainComplex

    CochainComplex(K):: PurePermComplex --> CochainComplex

    CochainComplex(Y):: RegCWComplex --> CochainComplex

    CochainComplex(K):: SimplicialComplex --> CohainComplex

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.

    CriticalCells(K):: RegCWComplex --> List

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.

    DiagonalApproximation(X):: RegCWComplex --> RegCWMap, RegCWMap

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.

    Size(Y):: RegCWComplex --> Int

    Size(Y):: SimplicialComplex --> Int

    Size(K):: PureCubicalComplex --> Int

    Size(K):: PurePermComplex --> Int

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.

(Co)chain Complexes ⟶ (Co)chain Complexes

    FilteredTensorWithIntegers(R):: FreeResolution, Int --> FilteredChainComplex

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 $\mathbb Z$.

    FilteredTensorWithIntegersModP(R,p):: FreeResolution, Int --> FilteredChainComplex

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 $\mathbb F$, the field of p elements.

    HomToIntegers(C):: ChainComplex --> CochainComplex

    HomToIntegers(R):: FreeResolution --> CochainComplex

    HomToIntegers(F):: EquiChainMap --> CochainMap

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).

    TensorWithIntegersModP(C,p):: ChainComplex, Int --> ChainComplex

    TensorWithIntegersModP(R,p):: FreeResolution, Int --> ChainComplex

    TensorWithIntegersModP(F,p):: EquiChainMap, Int --> ChainMap

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.

(Co)chain Complexes ⟶ Homotopy Invariants

    Cohomology(C,n):: CochainComplex, Int --> List

    Cohomology(F,n):: CochainMap, Int --> GroupHomomorphism

    Cohomology(K,n):: CubicalComplex, Int --> List

    Cohomology(K,n):: PureCubicalComplex, Int --> List

    Cohomology(K,n):: PurePermComplex, Int --> List

    Cohomology(K,n):: RegCWComplex, Int --> List

    Cohomology(K,n):: SimplicialComplex, Int --> List

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

    CupProduct(Y):: RegCWComplex --> Function

    CupProduct(R,p,q,P,Q):: FreeRes, Int, Int, List, List  --> List

Inputs a regular CW-complex Y and returns a function f(p,q,P,Q). This function f inputs two integers p,q ge 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 ge 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.

    Homology(C,n):: ChainComplex, Int --> List

    Homology(F,n):: ChainMap, Int --> GroupHomomorphism

    Homology(K,n):: CubicalComplex, Int --> List

    Homology(K,n):: PureCubicalComplex, Int --> List

    Homology(K,n):: PurePermComplex, Int --> List

    Homology(K,n):: RegCWComplex, Int --> List

    Homology(K,n):: SimplicialComplex, Int --> List

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

Visualization

    BarCodeDisplay(L) :: List --> void

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

    BarCodeCompactDisplay(L) :: List --> void

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

    CayleyGraphOfGroup(G,L):: Group, List --> Void

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.

    Display(G) :: Graph --> void

    Display(M) :: PureCubicalComplex --> void

    Display(M) :: PurePermutahedralComplex --> void

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

    DisplayArcPresentation(K) :: PureCubicalComplex --> void

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

    DisplayCSVKnotFile(str) :: String --> void

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

    DisplayDendrogram(L):: List --> Void

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

    DisplayDendrogramMat(A,t,s):: Mat, Rat, Int --> Void

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

    DisplayPDBfile(str):: String --> Void

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

    OrbitPolytope(G,v,L) :: PermGroup, List, List --> void

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.

    ScatterPlot(L):: List --> Void

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.

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