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
  (https://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832980).
  
  
  1.1 Data ⟶ Cellular Complexes
  
  1.1-1 RegularCWPolytope
  
  RegularCWPolytope( L )  function
  RegularCWPolytope( G, v )  function
  
  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
  
  CubicalComplex( A )  function
  
  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   (../tutorial/chap2.html) ,  2  (../tutorial/chap3.html) ,  3
  (../tutorial/chap5.html) ,       4       (../tutorial/chap10.html) ,       5
  (../www/SideLinks/About/aboutLinks.html) ,                                 6
  (../www/SideLinks/About/aboutPersistent.html) ,                            7
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        8
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                         9
  (../www/SideLinks/About/aboutCubical.html) ,                              10
  (../www/SideLinks/About/aboutRandomComplexes.html) ,                      11
  (../www/SideLinks/About/aboutTDA.html) ,                                  12
  (../www/SideLinks/About/aboutKnots.html) 
  
  1.1-3 PureCubicalComplex
  
  PureCubicalComplex( A )  function
  
  Inputs  a binary array A and returns the pure cubical complex represented by
  A.
  
  Examples:   1   (../tutorial/chap2.html) ,  2  (../tutorial/chap3.html) ,  3
  (../tutorial/chap5.html) ,       4       (../tutorial/chap10.html) ,       5
  (../www/SideLinks/About/aboutLinks.html) ,                                 6
  (../www/SideLinks/About/aboutPersistent.html) ,                            7
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        8
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                         9
  (../www/SideLinks/About/aboutCubical.html) ,                              10
  (../www/SideLinks/About/aboutRandomComplexes.html) ,                      11
  (../www/SideLinks/About/aboutTDA.html) ,                                  12
  (../www/SideLinks/About/aboutKnots.html) 
  
  1.1-4 PureCubicalKnot
  
  PureCubicalKnot( n, k )  function
  PureCubicalKnot( L )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,        4       (../tutorial/chap4.html) ,       5
  (../tutorial/chap6.html) ,                                                 6
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        7
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                         8
  (../www/SideLinks/About/aboutQuandles2.html) ,                             9
  (../www/SideLinks/About/aboutQuandles.html) ,                             10
  (../www/SideLinks/About/aboutKnots.html) ,                                11
  (../www/SideLinks/About/aboutKnotsQuandles.html) 
  
  1.1-5 PurePermutahedralKnot
  
  PurePermutahedralKnot( n, k )  function
  PurePermutahedralKnot( L )  function
  
  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 (../tutorial/chap1.html) , 2 (../tutorial/chap10.html) 
  
  1.1-6 PurePermutahedralComplex
  
  PurePermutahedralComplex( A )  function
  
  Inputs  a  binary  array  A  and  returns  the  pure  permutahedral  complex
  represented by A.
  
  Examples:   1   (../tutorial/chap2.html) ,  2  (../tutorial/chap5.html) ,  3
  (../www/SideLinks/About/aboutPeripheral.html) ,                            4
  (../www/SideLinks/About/aboutCubical.html) 
  
  1.1-7 CayleyGraphOfGroup
  
  CayleyGraphOfGroup( G, L )  function
  
  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
  
  EquivariantEuclideanSpace( G, v )  function
  
  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 (../tutorial/chap1.html) 
  
  1.1-9 EquivariantOrbitPolytope
  
  EquivariantOrbitPolytope( G, v )  function
  
  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
  
  EquivariantTwoComplex( G )  function
  
  Inputs  a  suitable group G and returns, as an equivariant regular CW-space,
  the 2-complex associated to some presentation of G.
  
  Examples: 1 (../tutorial/chap1.html) 
  
  1.1-11 QuillenComplex
  
  QuillenComplex( G, p )  function
  
  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  (../tutorial/chap1.html) ,  2  (../tutorial/chap10.html) ,  3
  (../www/SideLinks/About/aboutBredon.html) ,                                4
  (../www/SideLinks/About/aboutCubical.html) 
  
  1.1-12 RestrictedEquivariantCWComplex
  
  RestrictedEquivariantCWComplex( Y, H )  function
  
  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
  
  RandomSimplicialGraph( n, p )  function
  
  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 (../www/SideLinks/About/aboutRandomComplexes.html) 
  
  1.1-14 RandomSimplicialTwoComplex
  
  RandomSimplicialTwoComplex( n, p )  function
  
  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             (../tutorial/chap5.html) ,             2
  (../www/SideLinks/About/aboutRandomComplexes.html) 
  
  1.1-15 ReadCSVfileAsPureCubicalKnot
  
  ReadCSVfileAsPureCubicalKnot( str )  function
  ReadCSVfileAsPureCubicalKnot( str, r )  function
  ReadCSVfileAsPureCubicalKnot( L )  function
  ReadCSVfileAsPureCubicalKnot( L, R )  function
  
  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 (../tutorial/chap2.html) 
  
  1.1-16 ReadImageAsPureCubicalComplex
  
  ReadImageAsPureCubicalComplex( str, t )  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
  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  (../tutorial/chap5.html) ,  2  (../tutorial/chap10.html) ,  3
  (../www/SideLinks/About/aboutPersistent.html) ,                            4
  (../www/SideLinks/About/aboutCubical.html) ,                               5
  (../www/SideLinks/About/aboutTDA.html) 
  
  1.1-17 ReadImageAsFilteredPureCubicalComplex
  
  ReadImageAsFilteredPureCubicalComplex( str, n )  function
  
  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 (../tutorial/chap5.html) 
  
  1.1-18 ReadImageAsWeightFunction
  
  ReadImageAsWeightFunction( str, t )  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.
  
  Examples:
  
  1.1-19 ReadPDBfileAsPureCubicalComplex
  
  ReadPDBfileAsPureCubicalComplex( str )  function
  ReadPDBfileAsPureCubicalComplex( str, r )  function
  
  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             (../tutorial/chap5.html) ,             2
  (../www/SideLinks/About/aboutPersistent.html) ,                            3
  (../www/SideLinks/About/aboutKnots.html) 
  
  1.1-20 ReadPDBfileAsPurepermutahedralComplex
  
  ReadPDBfileAsPurepermutahedralComplex  global variable
  ReadPDBfileAsPurePermutahedralComplex( str, r )  function
  
  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
  
  RegularCWPolytope( L )  function
  RegularCWPolytope( G, v )  function
  
  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
  
  SimplicialComplex( L )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,        4       (../tutorial/chap4.html) ,       5
  (../tutorial/chap5.html) ,       6       (../tutorial/chap10.html) ,       7
  (../www/SideLinks/About/aboutMetrics.html) ,                               8
  (../www/SideLinks/About/aboutPersistent.html) ,                            9
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                       10
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                        11
  (../www/SideLinks/About/aboutCubical.html) ,                              12
  (../www/SideLinks/About/aboutRandomComplexes.html) 
  
  1.1-23 SymmetricMatrixToFilteredGraph
  
  SymmetricMatrixToFilteredGraph( A, m, s )  function
  SymmetricMatrixToFilteredGraph( A, m )  function
  
  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  (../tutorial/chap5.html) ,  2  (../tutorial/chap10.html) ,  3
  (../www/SideLinks/About/aboutPersistent.html) 
  
  1.1-24 SymmetricMatrixToGraph
  
  SymmetricMatrixToGraph( A, t )  function
  
  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             (../tutorial/chap5.html) ,             2
  (../www/SideLinks/About/aboutMetrics.html) 
  
  
  1.2 Metric Spaces
  
  1.2-1 CayleyMetric
  
  CayleyMetric( g, h )  function
  
  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 (../www/SideLinks/About/aboutMetrics.html) 
  
  1.2-2 EuclideanMetric
  
  EuclideanMetric  global variable
  
  Inputs two vectors v,w ∈ R^n and returns a rational number approximating the
  Euclidean distance between them.
  
  Examples:
  
  1.2-3 EuclideanSquaredMetric
  
  EuclideanSquaredMetric( g, h )  function
  
  Inputs  two  vectors  v,w  ∈  R^n  and  returns  the square of the Euclidean
  distance between them.
  
  Examples:
  
  1.2-4 HammingMetric
  
  HammingMetric( g, h )  function
  
  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
  
  KendallMetric( g, h )  function
  
  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
  
  ManhattanMetric( g, h )  function
  
  Inputs  two  vectors  v,w  ∈  R^n and returns the Manhattan distance between
  them.
  
  Examples: 1 (../www/SideLinks/About/aboutMetrics.html) 
  
  1.2-7 VectorsToSymmetricMatrix
  
  VectorsToSymmetricMatrix( V )  function
  VectorsToSymmetricMatrix( V, d )  function
  
  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  (../tutorial/chap5.html) ,  2  (../tutorial/chap10.html) ,  3
  (../www/SideLinks/About/aboutMetrics.html) 
  
  
  1.3 Cellular Complexes ⟶ Cellular Complexes
  
  1.3-1 BoundaryMap
  
  BoundaryMap( K )  function
  
  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  (../tutorial/chap2.html) ,  2  (../tutorial/chap10.html) ,  3
  (../www/SideLinks/About/aboutTopology.html) 
  
  1.3-2 CliqueComplex
  
  CliqueComplex( G, n )  function
  CliqueComplex( F, n )  function
  CliqueComplex( K, n )  function
  
  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 (../tutorial/chap5.html) 
  
  1.3-3 ConcentricFiltration
  
  ConcentricFiltration( K, n )  function
  
  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
  
  DirectProduct( M, N )  function
  DirectProduct( M, N )  function
  
  Inputs  two  or  more  regular  CW-complexes  or  two  or  more pure cubical
  complexes and returns their direct product.
  
  Examples:   1   (../tutorial/chap1.html) ,  2  (../tutorial/chap3.html) ,  3
  (../tutorial/chap10.html) ,                                                4
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        5
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                         6
  (../www/SideLinks/About/aboutCubical.html) ,                               7
  (../www/SideLinks/About/aboutExtensions.html) 
  
  1.3-5 FiltrationTerm
  
  FiltrationTerm( K, t )  function
  FiltrationTerm( K, t )  function
  
  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 (../tutorial/chap5.html) 
  
  1.3-6 Graph
  
  Graph( K )  function
  Graph( K )  function
  
  Inputs  a  regular  CW-complex  or  a  simplicial  complex K and returns its
  1-skeleton as a graph.
  
  Examples:   1   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap5.html) ,        4       (../tutorial/chap7.html) ,       5
  (../tutorial/chap10.html) ,       6       (../tutorial/chap11.html) ,      7
  (../www/SideLinks/About/aboutMetrics.html) ,                               8
  (../www/SideLinks/About/aboutPersistent.html) ,                            9
  (../www/SideLinks/About/aboutRandomComplexes.html) ,                      10
  (../www/SideLinks/About/aboutSpaceGroup.html) ,                           11
  (../www/SideLinks/About/aboutGraphsOfGroups.html) ,                       12
  (../www/SideLinks/About/aboutIntro.html) ,                                13
  (../www/SideLinks/About/aboutTopology.html) ,                             14
  (../www/SideLinks/About/aboutTwistedCoefficients.html) 
  
  1.3-7 HomotopyGraph
  
  HomotopyGraph( Y )  function
  
  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 (../tutorial/chap5.html) 
  
  1.3-8 Nerve
  
  Nerve( M )  function
  Nerve( M )  function
  Nerve( M, n )  function
  Nerve( M, n )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap10.html) ,       4       (../tutorial/chap12.html) ,      5
  (../www/SideLinks/About/aboutMetrics.html) ,                               6
  (../www/SideLinks/About/aboutPersistent.html) ,                            7
  (../www/SideLinks/About/aboutRandomComplexes.html) ,                       8
  (../www/SideLinks/About/aboutSimplicialGroups.html) ,                      9
  (../www/SideLinks/About/aboutIntro.html) 
  
  1.3-9 RegularCWComplex
  
  RegularCWComplex( K )  function
  RegularCWComplex( K )  function
  RegularCWComplex( K )  function
  RegularCWComplex( K )  function
  RegularCWComplex( L )  function
  RegularCWComplex( L, M )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,        4       (../tutorial/chap4.html) ,       5
  (../tutorial/chap5.html) ,       6       (../tutorial/chap10.html) ,       7
  (../www/SideLinks/About/aboutMetrics.html) ,                               8
  (../www/SideLinks/About/aboutPeripheral.html) ,                            9
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                       10
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                        11
  (../www/SideLinks/About/aboutRandomComplexes.html) ,                      12
  (../www/SideLinks/About/aboutKnots.html) 
  
  1.3-10 RegularCWMap
  
  RegularCWMap( M, A )  function
  
  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             (../tutorial/chap4.html) ,             2
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        3
  (../www/SideLinks/About/aboutCoverinSpaces.html) 
  
  1.3-11 ThickeningFiltration
  
  ThickeningFiltration( K, n )  function
  ThickeningFiltration( K, n, s )  function
  
  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             (../tutorial/chap5.html) ,             2
  (../www/SideLinks/About/aboutPersistent.html) 
  
  
  1.4 Cellular Complexes ⟶ Cellular Complexes (Preserving Data Types)
  
  1.4-1 ContractedComplex
  
  ContractedComplex( K )  function
  ContractedComplex( K )  function
  ContractedComplex( K )  function
  ContractedComplex( K )  function
  ContractedComplex( K, S )  function
  ContractedComplex( K )  function
  ContractedComplex( K )  function
  ContractedComplex( K, S )  function
  ContractedComplex( K )  function
  ContractedComplex( G )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,        4       (../tutorial/chap5.html) ,       5
  (../tutorial/chap7.html) ,       6       (../tutorial/chap10.html) ,       7
  (../tutorial/chap11.html) ,                                                8
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        9
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                        10
  (../www/SideLinks/About/aboutCubical.html) ,                              11
  (../www/SideLinks/About/aboutKnots.html) 
  
  1.4-2 ContractibleSubcomplex
  
  ContractibleSubcomplex( K )  function
  ContractibleSubcomplex( K )  function
  ContractibleSubcomplex( K )  function
  
  Inputs  a non-empty pure cubical, pure permutahedral or simplicial complex K
  and returns a contractible subcomplex.
  
  Examples:             1             (../tutorial/chap10.html) ,            2
  (../www/SideLinks/About/aboutCubical.html) 
  
  1.4-3 KnotReflection
  
  KnotReflection( K )  function
  
  Inputs a pure cubical knot and returns the reflected knot.
  
  Examples:
  
  1.4-4 KnotSum
  
  KnotSum( K, L )  function
  
  Inputs two pure cubical knots and returns their sum.
  
  Examples:   1   (../tutorial/chap2.html) ,  2  (../tutorial/chap3.html) ,  3
  (../tutorial/chap6.html) ,                                                 4
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                         5
  (../www/SideLinks/About/aboutKnots.html) 
  
  1.4-5 OrientRegularCWComplex
  
  OrientRegularCWComplex( Y )  function
  
  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
  
  PathComponent( K, n )  function
  PathComponent( K, n )  function
  PathComponent( K, n )  function
  
  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             (../tutorial/chap5.html) ,             2
  (../www/SideLinks/About/aboutQuandles.html) ,                              3
  (../www/SideLinks/About/aboutTDA.html) 
  
  1.4-7 PureComplexBoundary
  
  PureComplexBoundary( M )  function
  PureComplexBoundary( M )  function
  
  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 (../tutorial/chap5.html) 
  
  1.4-8 PureComplexComplement
  
  PureComplexComplement( M )  function
  PureComplexComplement( M )  function
  
  Inputs  a  pure  cubical complex or a pure permutahedral complex and returns
  its complement.
  
  Examples:   1   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,        4       (../tutorial/chap5.html) ,       5
  (../tutorial/chap10.html) ,                                                6
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        7
  (../www/SideLinks/About/aboutCoverinSpaces.html) 
  
  1.4-9 PureComplexDifference
  
  PureComplexDifference( M, N )  function
  PureComplexDifference( M, N )  function
  
  Inputs  two  pure  cubical complexes or two pure permutahedral complexes and
  returns the difference M - N.
  
  Examples: 1 (../tutorial/chap5.html) 
  
  1.4-10 PureComplexInterstection
  
  PureComplexInterstection  global variable
  PureComplexIntersection( M, N )  function
  
  Inputs  two  pure  cubical complexes or two pure permutahedral complexes and
  returns their intersection.
  
  Examples:
  
  1.4-11 PureComplexThickened
  
  PureComplexThickened( M )  function
  PureComplexThickened( M )  function
  
  Inputs  a  pure  cubical complex or a pure permutahedral complex and returns
  the a thickened complex.
  
  Examples: 1 (../tutorial/chap5.html) 
  
  1.4-12 PureComplexUnion
  
  PureComplexUnion( M, N )  function
  PureComplexUnion( M, N )  function
  
  Inputs  two  pure  cubical complexes or two pure permutahedral complexes and
  returns their union.
  
  Examples: 1 (../tutorial/chap5.html) 
  
  1.4-13 SimplifiedComplex
  
  SimplifiedComplex( K )  function
  SimplifiedComplex( K )  function
  SimplifiedComplex( R )  function
  SimplifiedComplex( C )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap3.html) ,  3
  (../tutorial/chap4.html) ,       4       (../tutorial/chap11.html) ,       5
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        6
  (../www/SideLinks/About/aboutCoverinSpaces.html) 
  
  1.4-14 ZigZagContractedComplex
  
  ZigZagContractedComplex( K )  function
  ZigZagContractedComplex( K )  function
  ZigZagContractedComplex( K )  function
  
  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 (../tutorial/chap2.html) 
  
  
  1.5 Cellular Complexes ⟶ Homotopy Invariants
  
  1.5-1 AlexanderPolynomial
  
  AlexanderPolynomial( K )  function
  AlexanderPolynomial( K )  function
  AlexanderPolynomial( G )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap5.html) , 4 (../www/SideLinks/About/aboutKnots.html) 
  
  1.5-2 BettiNumber
  
  BettiNumber( K, n )  function
  BettiNumber( K, n )  function
  BettiNumber( K, n )  function
  BettiNumber( K, n )  function
  BettiNumber( K, n )  function
  BettiNumber( K, n )  function
  BettiNumber( K, n )  function
  BettiNumber( K, n, p )  function
  BettiNumber( K, n, p )  function
  BettiNumber( K, n, p )  function
  BettiNumber( K, n, p )  function
  BettiNumber( K, n, p )  function
  
  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 (../tutorial/chap5.html) 
  
  1.5-3 EulerCharacteristic
  
  EulerCharacteristic( C )  function
  EulerCharacteristic( K )  function
  EulerCharacteristic( K )  function
  EulerCharacteristic( K )  function
  EulerCharacteristic( K )  function
  EulerCharacteristic( K )  function
  
  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
  
  EulerIntegral( Y, w )  function
  
  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
  
  FundamentalGroup( K )  function
  FundamentalGroup( K, n )  function
  FundamentalGroup( K )  function
  FundamentalGroup( K )  function
  FundamentalGroup( K )  function
  FundamentalGroup( F )  function
  FundamentalGroup( F, n )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,        4       (../tutorial/chap4.html) ,       5
  (../tutorial/chap5.html) ,       6       (../tutorial/chap11.html) ,       7
  (../www/SideLinks/About/aboutLinks.html) ,                                 8
  (../www/SideLinks/About/aboutPeripheral.html) ,                            9
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                       10
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                        11
  (../www/SideLinks/About/aboutQuandles.html) ,                             12
  (../www/SideLinks/About/aboutRandomComplexes.html) ,                      13
  (../www/SideLinks/About/aboutKnots.html) 
  
  1.5-6 FundamentalGroupOfQuotient
  
  FundamentalGroupOfQuotient( Y )  function
  
  Inputs a G-equivariant regular CW complex Y and returns the group G.
  
  Examples: 1 (../tutorial/chap1.html) 
  
  1.5-7 IsAspherical
  
  IsAspherical( F, R )  function
  
  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   (../tutorial/chap3.html) ,  2  (../tutorial/chap6.html) ,  3
  (../www/SideLinks/About/aboutAspherical.html) ,                            4
  (../www/SideLinks/About/aboutIntro.html) 
  
  1.5-8 KnotGroup
  
  KnotGroup( K )  function
  KnotGroup( K )  function
  
  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 (../www/SideLinks/About/aboutKnots.html) 
  
  1.5-9 PiZero
  
  PiZero( Y )  function
  PiZero( Y )  function
  PiZero( Y )  function
  
  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 (../tutorial/chap5.html) 
  
  1.5-10 PersistentBettiNumbers
  
  PersistentBettiNumbers( K, n )  function
  PersistentBettiNumbers( K, n )  function
  PersistentBettiNumbers( K, n )  function
  PersistentBettiNumbers( K, n )  function
  PersistentBettiNumbers( K, n )  function
  PersistentBettiNumbers( K, n, p )  function
  PersistentBettiNumbers( K, n, p )  function
  PersistentBettiNumbers( K, n, p )  function
  PersistentBettiNumbers( K, n, p )  function
  PersistentBettiNumbers( K, n, p )  function
  
  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 (../tutorial/chap5.html) 
  
  
  1.6 Data ⟶ Homotopy Invariants
  
  1.6-1 DendrogramMat
  
  DendrogramMat( A, t, s )  function
  
  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
  
  ChainComplex( K )  function
  ChainComplex( K )  function
  ChainComplex( K )  function
  ChainComplex( Y )  function
  ChainComplex( K )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap3.html) ,  3
  (../tutorial/chap4.html) ,       4       (../tutorial/chap10.html) ,       5
  (../tutorial/chap12.html) , 6 (../www/SideLinks/About/aboutMetrics.html) , 7
  (../www/SideLinks/About/aboutBredon.html) ,                                8
  (../www/SideLinks/About/aboutPersistent.html) ,                            9
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                       10
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                        11
  (../www/SideLinks/About/aboutCubical.html) ,                              12
  (../www/SideLinks/About/aboutSimplicialGroups.html) ,                     13
  (../www/SideLinks/About/aboutIntro.html) 
  
  1.7-2 ChainComplexEquivalence
  
  ChainComplexEquivalence  global variable
  
  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
  
  ChainComplexOfQuotient( Y )  function
  
  Inputs  a  G-equivariant regular CW-complex Y and returns the cellular chain
  complex of the quotient space Y/G.
  
  Examples: 1 (../tutorial/chap1.html) 
  
  1.7-4 ChainMap
  
  ChainMap( X, A, Y, B )  function
  ChainMap( f )  function
  ChainMap( f )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap7.html) ,  3
  (../tutorial/chap10.html) ,                                                4
  (../www/SideLinks/About/aboutCohomologyRings.html) ,                       5
  (../www/SideLinks/About/aboutPoincareSeries.html) ,                        6
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                        7
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                         8
  (../www/SideLinks/About/aboutFunctorial.html) 
  
  1.7-5 CochainComplex
  
  CochainComplex( K )  function
  CochainComplex( K )  function
  CochainComplex( K )  function
  CochainComplex( Y )  function
  CochainComplex( K )  function
  
  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
  
  CriticalCells( K )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,  4  (../www/SideLinks/About/aboutLinks.html) ,  5
  (../www/SideLinks/About/aboutPeripheral.html) ,                            6
  (../www/SideLinks/About/aboutCubical.html) ,                               7
  (../www/SideLinks/About/aboutRandomComplexes.html) ,                       8
  (../www/SideLinks/About/aboutKnots.html) 
  
  1.7-7 DiagonalApproximation
  
  DiagonalApproximation( X )  function
  
  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 (../tutorial/chap1.html) 
  
  1.7-8 Size
  
  Size( Y )  function
  Size( Y )  function
  Size( K )  function
  Size( K )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,        4       (../tutorial/chap4.html) ,       5
  (../tutorial/chap5.html) ,        6       (../tutorial/chap7.html) ,       7
  (../tutorial/chap10.html) ,       8       (../tutorial/chap11.html) ,      9
  (../tutorial/chap12.html) , 10 (../www/SideLinks/About/aboutLinks.html) , 11
  (../www/SideLinks/About/aboutMetrics.html) ,                              12
  (../www/SideLinks/About/aboutCoefficientSequence.html) ,                  13
  (../www/SideLinks/About/aboutPeripheral.html) ,                           14
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                       15
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                        16
  (../www/SideLinks/About/aboutQuandles2.html) ,                            17
  (../www/SideLinks/About/aboutQuandles.html) ,                             18
  (../www/SideLinks/About/aboutCubical.html) ,                              19
  (../www/SideLinks/About/aboutSimplicialGroups.html) ,                     20
  (../www/SideLinks/About/aboutTDA.html) ,                                  21
  (../www/SideLinks/About/aboutKnots.html) 
  
  
  1.8 (Co)chain Complexes ⟶ (Co)chain Complexes
  
  1.8-1 FilteredTensorWithIntegers
  
  FilteredTensorWithIntegers( R )  function
  
  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             (../tutorial/chap10.html) ,            2
  (../www/SideLinks/About/aboutPersistent.html) 
  
  1.8-2 FilteredTensorWithIntegersModP
  
  FilteredTensorWithIntegersModP( R, p )  function
  
  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             (../tutorial/chap10.html) ,            2
  (../www/SideLinks/About/aboutPersistent.html) 
  
  1.8-3 HomToIntegers
  
  HomToIntegers( C )  function
  HomToIntegers( R )  function
  HomToIntegers( F )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap7.html) ,  3
  (../tutorial/chap8.html) ,       4       (../tutorial/chap10.html) ,       5
  (../tutorial/chap13.html) ,                                                6
  (../www/SideLinks/About/aboutCohomologyRings.html) ,                       7
  (../www/SideLinks/About/aboutSpaceGroup.html) ,                            8
  (../www/SideLinks/About/aboutIntro.html) ,                                 9
  (../www/SideLinks/About/aboutTorAndExt.html) 
  
  1.8-4 TensorWithIntegersModP
  
  TensorWithIntegersModP( C, p )  function
  TensorWithIntegersModP( R, p )  function
  TensorWithIntegersModP( F, p )  function
  
  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  (../tutorial/chap1.html) ,  2  (../tutorial/chap10.html) ,  3
  (../www/SideLinks/About/aboutArithmetic.html) ,                            4
  (../www/SideLinks/About/aboutPerformance.html) ,                           5
  (../www/SideLinks/About/aboutPersistent.html) ,                            6
  (../www/SideLinks/About/aboutPoincareSeries.html) ,                        7
  (../www/SideLinks/About/aboutDefinitions.html) ,                           8
  (../www/SideLinks/About/aboutExtensions.html) ,                            9
  (../www/SideLinks/About/aboutTorAndExt.html) 
  
  
  1.9 (Co)chain Complexes ⟶ Homotopy Invariants
  
  1.9-1 Cohomology
  
  Cohomology( C, n )  function
  Cohomology( F, n )  function
  Cohomology( K, n )  function
  Cohomology( K, n )  function
  Cohomology( K, n )  function
  Cohomology( K, n )  function
  Cohomology( K, n )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap3.html) ,  3
  (../tutorial/chap4.html) ,        4       (../tutorial/chap6.html) ,       5
  (../tutorial/chap7.html) ,        6       (../tutorial/chap8.html) ,       7
  (../tutorial/chap12.html) ,       8       (../tutorial/chap13.html) ,      9
  (../tutorial/chap14.html) ,                                               10
  (../www/SideLinks/About/aboutArtinGroups.html) ,                          11
  (../www/SideLinks/About/aboutModPRings.html) ,                            12
  (../www/SideLinks/About/aboutNoncrossing.html) ,                          13
  (../www/SideLinks/About/aboutCoefficientSequence.html) ,                  14
  (../www/SideLinks/About/aboutCohomologyRings.html) ,                      15
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                       16
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                        17
  (../www/SideLinks/About/aboutCoxeter.html) ,                              18
  (../www/SideLinks/About/aboutCrossedMods.html) ,                          19
  (../www/SideLinks/About/aboutExtensions.html) ,                           20
  (../www/SideLinks/About/aboutSpaceGroup.html) ,                           21
  (../www/SideLinks/About/aboutGouter.html) ,                               22
  (../www/SideLinks/About/aboutSurvey.html) ,                               23
  (../www/SideLinks/About/aboutIntro.html) ,                                24
  (../www/SideLinks/About/aboutTopology.html) ,                             25
  (../www/SideLinks/About/aboutTorAndExt.html) ,                            26
  (../www/SideLinks/About/aboutTwistedCoefficients.html) 
  
  1.9-2 CupProduct
  
  CupProduct( Y )  function
  CupProduct( R, p, q, P, Q )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap5.html) ,        4       (../tutorial/chap7.html) ,       5
  (../www/SideLinks/About/aboutCohomologyRings.html) 
  
  1.9-3 Homology
  
  Homology( C, n )  function
  Homology( F, n )  function
  Homology( K, n )  function
  Homology( K, n )  function
  Homology( K, n )  function
  Homology( K, n )  function
  Homology( K, n )  function
  
  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   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap3.html) ,        4       (../tutorial/chap4.html) ,       5
  (../tutorial/chap5.html) ,        6       (../tutorial/chap7.html) ,       7
  (../tutorial/chap9.html) ,       8       (../tutorial/chap10.html) ,       9
  (../tutorial/chap11.html) ,      10      (../tutorial/chap12.html) ,      11
  (../tutorial/chap13.html) , 12 (../www/SideLinks/About/aboutLinks.html) , 13
  (../www/SideLinks/About/aboutArithmetic.html) ,                           14
  (../www/SideLinks/About/aboutMetrics.html) ,                              15
  (../www/SideLinks/About/aboutArtinGroups.html) ,                          16
  (../www/SideLinks/About/aboutAspherical.html) ,                           17
  (../www/SideLinks/About/aboutParallel.html) ,                             18
  (../www/SideLinks/About/aboutBredon.html) ,                               19
  (../www/SideLinks/About/aboutPerformance.html) ,                          20
  (../www/SideLinks/About/aboutCocycles.html) ,                             21
  (../www/SideLinks/About/aboutPersistent.html) ,                           22
  (../www/SideLinks/About/aboutPoincareSeries.html) ,                       23
  (../www/SideLinks/About/aboutCoveringSpaces.html) ,                       24
  (../www/SideLinks/About/aboutCoverinSpaces.html) ,                        25
  (../www/SideLinks/About/aboutPolytopes.html) ,                            26
  (../www/SideLinks/About/aboutCoxeter.html) ,                              27
  (../www/SideLinks/About/aboutquasi.html) ,                                28
  (../www/SideLinks/About/aboutCubical.html) ,                              29
  (../www/SideLinks/About/aboutRandomComplexes.html) ,                      30
  (../www/SideLinks/About/aboutRosenbergerMonster.html) ,                   31
  (../www/SideLinks/About/aboutDavisComplex.html) ,                         32
  (../www/SideLinks/About/aboutDefinitions.html) ,                          33
  (../www/SideLinks/About/aboutSimplicialGroups.html) ,                     34
  (../www/SideLinks/About/aboutExtensions.html) ,                           35
  (../www/SideLinks/About/aboutSpaceGroup.html) ,                           36
  (../www/SideLinks/About/aboutFunctorial.html) ,                           37
  (../www/SideLinks/About/aboutGraphsOfGroups.html) ,                       38
  (../www/SideLinks/About/aboutIntro.html) ,                                39
  (../www/SideLinks/About/aboutTensorSquare.html) ,                         40
  (../www/SideLinks/About/aboutLieCovers.html) ,                            41
  (../www/SideLinks/About/aboutTorAndExt.html) ,                            42
  (../www/SideLinks/About/aboutLie.html) ,                                  43
  (../www/SideLinks/About/aboutTwistedCoefficients.html) 
  
  
  1.10 Visualization
  
  1.10-1 BarCodeDisplay
  
  BarCodeDisplay( L )  function
  
  Displays a barcode L=PersitentBettiNumbers(X,n).
  
  Examples:             1             (../tutorial/chap10.html) ,            2
  (../www/SideLinks/About/aboutPersistent.html) 
  
  1.10-2 BarCodeCompactDisplay
  
  BarCodeCompactDisplay( L )  function
  
  Displays a barcode L=PersitentBettiNumbers(X,n) in compact form.
  
  Examples:   1  (../tutorial/chap5.html) ,  2  (../tutorial/chap10.html) ,  3
  (../www/SideLinks/About/aboutPersistent.html) 
  
  1.10-3 CayleyGraphOfGroup
  
  CayleyGraphOfGroup( G, L )  function
  
  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
  
  Display( G )  function
  Display( M )  function
  Display( M )  function
  
  Displays  a  graph  G;  a  2-  or  3-dimensional  pure  cubical complex M; a
  3-dimensional pure permutahedral complex M.
  
  Examples:   1   (../tutorial/chap1.html) ,  2  (../tutorial/chap2.html) ,  3
  (../tutorial/chap4.html) ,        4       (../tutorial/chap5.html) ,       5
  (../tutorial/chap6.html) ,        6       (../tutorial/chap7.html) ,       7
  (../tutorial/chap9.html) ,       8       (../tutorial/chap10.html) ,       9
  (../tutorial/chap11.html) ,      10      (../tutorial/chap13.html) ,      11
  (../tutorial/chap14.html) ,  12 (../www/SideLinks/About/aboutMetrics.html) ,
  13            (../www/SideLinks/About/aboutArtinGroups.html) ,            14
  (../www/SideLinks/About/aboutNoncrossing.html) ,                          15
  (../www/SideLinks/About/aboutPeriodic.html) ,                             16
  (../www/SideLinks/About/aboutPersistent.html) ,                           17
  (../www/SideLinks/About/aboutPolytopes.html) ,                            18
  (../www/SideLinks/About/aboutQuandles2.html) ,                            19
  (../www/SideLinks/About/aboutQuandles.html) ,                             20
  (../www/SideLinks/About/aboutSuperperfect.html) ,                         21
  (../www/SideLinks/About/aboutGraphsOfGroups.html) ,                       22
  (../www/SideLinks/About/aboutIntro.html) ,                                23
  (../www/SideLinks/About/aboutKnotsQuandles.html) ,                        24
  (../www/SideLinks/About/aboutTopology.html) 
  
  1.10-5 DisplayArcPresentation
  
  DisplayArcPresentation( K )  function
  
  Displays  a 3-dimensional pure cubical knot K=PureCubicalKnot(L) in the form
  of an arc presentation.
  
  Examples:
  
  1.10-6 DisplayCSVKnotFile
  
  DisplayCSVKnotFile  global variable
  
  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
  
  DisplayDendrogram( L )  function
  
  Displays the dendrogram L:=DendrogramMat(A,t,s).
  
  Examples:
  
  1.10-8 DisplayDendrogramMat
  
  DisplayDendrogramMat( A, t, s )  function
  
  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
  
  DisplayPDBfile( str )  function
  
  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 (../tutorial/chap5.html) 
  
  1.10-10 OrbitPolytope
  
  OrbitPolytope( G, v, L )  function
  
  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             (../tutorial/chap11.html) ,            2
  (../www/SideLinks/About/aboutPolytopes.html) 
  
  1.10-11 ScatterPlot
  
  ScatterPlot( L )  function
  
  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 (../tutorial/chap5.html)