Interface to CHomP

CHomP stands for “Computation Homology Program”, and is good at computing homology of simplicial complexes, cubical complexes, and chain complexes. It can also compute homomorphisms induced on homology by maps. See the CHomP web page http://chomp.rutgers.edu/ for more information.

AUTHOR:

  • John H. Palmieri

class sage.interfaces.chomp.CHomP

Bases: object

Interface to the CHomP package.

Parameters

  • program (string) – which CHomP program to use

  • complex – a simplicial or cubical complex

  • subcomplex – a subcomplex of complex or None (the default)

  • base_ring (ring; optional, default \(\ZZ\)) – ring over which to perform computations – must be \(\ZZ\) or \(\GF{p}\).

  • generators (boolean; optional, default False) – if True, also return list of generators

  • verbose (boolean; optional, default False) – if True, print helpful messages as the computation progresses

  • extra_opts (string) – options passed directly to program

Returns

homology groups as a dictionary indexed by dimension

The programs homsimpl, homcubes, and homchain are available through this interface. homsimpl computes the relative or absolute homology groups of simplicial complexes. homcubes computes the relative or absolute homology groups of cubical complexes. homchain computes the homology groups of chain complexes. For consistency with Sage’s other homology computations, the answers produced by homsimpl and homcubes in the absolute case are converted to reduced homology.

Note also that CHomP can only compute over the integers or \(\GF{p}\). CHomP is fast enough, though, that if you want rational information, you should consider using CHomP with integer coefficients, or with mod \(p\) coefficients for a sufficiently large \(p\), rather than using Sage’s built-in homology algorithms.

See also the documentation for the functions homchain(), homcubes(), and homsimpl() for more examples, including illustrations of some of the optional parameters.

EXAMPLES:

sage: from sage.interfaces.chomp import CHomP
sage: T = cubical_complexes.Torus()
sage: CHomP()('homcubes', T) # optional - CHomP
{0: 0, 1: Z x Z, 2: Z}

Relative homology of a segment relative to its endpoints:

sage: edge = simplicial_complexes.Simplex(1)
sage: ends = edge.n_skeleton(0)
sage: CHomP()('homsimpl', edge)  # optional - CHomP
{0: 0}
sage: CHomP()('homsimpl', edge, ends)  # optional - CHomP
{0: 0, 1: Z}

Homology of a chain complex:

sage: C = ChainComplex({3: 2 * identity_matrix(ZZ, 2)}, degree=-1)
sage: CHomP()('homchain', C)  # optional - CHomP
{2: C2 x C2}
help(program)

Print a help message for program, a program from the CHomP suite.

Parameters

program (string) – which CHomP program to use

Returns

nothing – just print a message

EXAMPLES:

sage: from sage.interfaces.chomp import CHomP
sage: CHomP().help('homcubes')   # optional - CHomP
HOMCUBES, ver. ... Copyright (C) ... by Pawel Pilarczyk...
sage.interfaces.chomp.have_chomp(program='homsimpl')

Return True if this computer has program installed.

The first time it is run, this function caches its result in the variable _have_chomp – a dictionary indexed by program name – and any subsequent time, it just checks the value of the variable.

This program is used in the routine CHomP.__call__.

If this computer doesn’t have CHomP installed, you may obtain it from http://chomp.rutgers.edu/.

EXAMPLES:

sage: from sage.interfaces.chomp import have_chomp
sage: have_chomp() # random -- depends on whether CHomP is installed
True
sage: 'homsimpl' in sage.interfaces.chomp._have_chomp
True
sage: sage.interfaces.chomp._have_chomp['homsimpl'] == have_chomp()
True
sage.interfaces.chomp.homchain(complex=None, **kwds)

Compute the homology of a chain complex using the CHomP program homchain.

Parameters

  • complex – a chain complex

  • generators (boolean; optional, default False) – if True, also return list of generators

  • verbose (boolean; optional, default False) – if True, print helpful messages as the computation progresses

  • help (boolean; optional, default False) – if True, just print a help message and exit

  • extra_opts (string) – options passed directly to homchain

Returns

homology groups as a dictionary indexed by dimension

EXAMPLES:

sage: from sage.interfaces.chomp import homchain
sage: C = cubical_complexes.Sphere(3).chain_complex()
sage: homchain(C)[3]   # optional - CHomP
Z

Generators: these are given as a list after the homology group. Each generator is specified as a cycle, an element in the appropriate free module over the base ring:

sage: C2 = delta_complexes.Sphere(2).chain_complex()
sage: homchain(C2, generators=True)[2]  # optional - CHomP
(Z, [(1, -1)])
sage: homchain(C2, generators=True, base_ring=GF(2))[2]  # optional - CHomP
(Vector space of dimension 1 over Finite Field of size 2, [(1, 1)])
sage.interfaces.chomp.homcubes(complex=None, subcomplex=None, **kwds)

Compute the homology of a cubical complex using the CHomP program homcubes. If the argument subcomplex is present, compute homology of complex relative to subcomplex.

Parameters

  • complex – a cubical complex

  • subcomplex – a subcomplex of complex or None (the default)

  • base_ring (ring; optional, default \(\ZZ\)) – ring over which to perform computations – must be \(\ZZ\) or \(\GF{p}\).

  • generators (boolean; optional, default False) – if True, also return list of generators

  • verbose (boolean; optional, default False) – if True, print helpful messages as the computation progresses

  • help (boolean; optional, default False) – if True, just print a help message and exit

  • extra_opts (string) – options passed directly to homcubes

Returns

homology groups as a dictionary indexed by dimension

EXAMPLES:

sage: from sage.interfaces.chomp import homcubes
sage: S = cubical_complexes.Sphere(3)
sage: homcubes(S)[3]   # optional - CHomP
Z

Relative homology:

sage: C3 = cubical_complexes.Cube(3)
sage: bdry = C3.n_skeleton(2)
sage: homcubes(C3, bdry)   # optional - CHomP
{0: 0, 1: 0, 2: 0, 3: Z}

Generators: these are given as a list after the homology group. Each generator is specified as a linear combination of cubes:

sage: homcubes(cubical_complexes.Sphere(1), generators=True, base_ring=GF(2))[1][1]   # optional - CHomP
[[[1,1] x [0,1]] + [[0,1] x [1,1]] + [[0,1] x [0,0]] + [[0,0] x [0,1]]]
sage.interfaces.chomp.homsimpl(complex=None, subcomplex=None, **kwds)

Compute the homology of a simplicial complex using the CHomP program homsimpl. If the argument subcomplex is present, compute homology of complex relative to subcomplex.

Parameters

  • complex – a simplicial complex

  • subcomplex – a subcomplex of complex or None (the default)

  • base_ring (ring; optional, default \(\ZZ\)) – ring over which to perform computations – must be \(\ZZ\) or \(\GF{p}\).

  • generators (boolean; optional, default False) – if True, also return list of generators

  • verbose (boolean; optional, default False) – if True, print helpful messages as the computation progresses

  • help (boolean; optional, default False) – if True, just print a help message and exit

  • extra_opts (string) – options passed directly to program

Returns

homology groups as a dictionary indexed by dimension

EXAMPLES:

sage: from sage.interfaces.chomp import homsimpl
sage: T = simplicial_complexes.Torus()
sage: M8 = simplicial_complexes.MooreSpace(8)
sage: M4 = simplicial_complexes.MooreSpace(4)
sage: X = T.disjoint_union(T).disjoint_union(T).disjoint_union(M8).disjoint_union(M4)
sage: homsimpl(X)[1]  # optional - CHomP
Z^6 x C4 x C8

Relative homology:

sage: S = simplicial_complexes.Simplex(3)
sage: bdry = S.n_skeleton(2)
sage: homsimpl(S, bdry)[3]   # optional - CHomP
Z

Generators: these are given as a list after the homology group. Each generator is specified as a linear combination of simplices:

sage: homsimpl(S, bdry, generators=True)[3]   # optional - CHomP
(Z, [(0, 1, 2, 3)])

sage: homsimpl(simplicial_complexes.Sphere(1), generators=True)   # optional - CHomP
{0: 0, 1: (Z, [(0, 1) - (0, 2) + (1, 2)])}
sage.interfaces.chomp.process_generators_chain(gen_string, dim, base_ring=None)

Process CHomP generator information for simplicial complexes.

Parameters

  • gen_string (string) – generator output from CHomP

  • dim (integer) – dimension in which to find generators

  • base_ring (optional, default ZZ) – base ring over which to do the computations

Returns

list of generators in each dimension, as described below

gen_string has the form

[H_0]
a1

[H_1]
a2
a3

[H_2]
a1 - a2

For each homology group, each line lists a homology generator as a linear combination of generators ai of the group of chains in the appropriate dimension. The elements ai are indexed starting with \(i=1\). Each generator is converted, using regular expressions, from a string to a vector (an element in the free module over base_ring), with ai representing the unit vector in coordinate \(i-1\). For example, the string a1 - a2 gets converted to the vector (1, -1).

Therefore the return value is a list of vectors.

EXAMPLES:

sage: from sage.interfaces.chomp import process_generators_chain
sage: s = "[H_0]\na1\n\n[H_1]\na2\na3\n"
sage: process_generators_chain(s, 1)
[(0, 1), (0, 0, 1)]
sage: s = "[H_0]\na1\n\n[H_1]\n5 * a2 - a1\na3\n"
sage: process_generators_chain(s, 1, base_ring=ZZ)
[(-1, 5), (0, 0, 1)]
sage: process_generators_chain(s, 1, base_ring=GF(2))
[(1, 1), (0, 0, 1)]
sage.interfaces.chomp.process_generators_cubical(gen_string, dim)

Process CHomP generator information for cubical complexes.

Parameters

  • gen_string (string) – generator output from CHomP

  • dim (integer) – dimension in which to find generators

Returns

list of generators in each dimension, as described below

gen_string has the form

The 2 generators of H_1 follow:
generator 1
-1 * [(0,0,0,0,0)(0,0,0,0,1)]
1 * [(0,0,0,0,0)(0,0,1,0,0)]
...
generator 2
-1 * [(0,1,0,1,1)(1,1,0,1,1)]
-1 * [(0,1,0,0,1)(0,1,0,1,1)]
...

Each line consists of a coefficient multiplied by a cube; the cube is specified by its “bottom left” and “upper right” corners.

For technical reasons, we remove the first coordinate of each tuple, and using regular expressions, the remaining parts get converted from a string to a pair (coefficient, Cube), with the cube represented as a product of tuples. For example, the first line in “generator 1” gets turned into

(-1, [0,0] x [0,0] x [0,0] x [0,1])

representing an element in the free abelian group with basis given by cubes. Each generator is a list of such pairs, representing the sum of such elements. These are reassembled in CHomP.__call__() to actual elements in the free module generated by the cubes of the cubical complex in the appropriate dimension.

Therefore the return value is a list of lists of pairs, one list of pairs for each generator.

EXAMPLES:

sage: from sage.interfaces.chomp import process_generators_cubical
sage: s = "The 2 generators of H_1 follow:\ngenerator 1:\n-1 * [(0,0,0,0,0)(0,0,0,0,1)]\n1 * [(0,0,0,0,0)(0,0,1,0,0)]"
sage: process_generators_cubical(s, 1)
[[(-1, [0,0] x [0,0] x [0,0] x [0,1]), (1, [0,0] x [0,1] x [0,0] x [0,0])]]
sage: len(process_generators_cubical(s, 1))  # only one generator
1
sage.interfaces.chomp.process_generators_simplicial(gen_string, dim, complex)

Process CHomP generator information for simplicial complexes.

Parameters

  • gen_string (string) – generator output from CHomP

  • dim (integer) – dimension in which to find generators

  • complex – simplicial complex under consideration

Returns

list of generators in each dimension, as described below

gen_string has the form

The 2 generators of H_1 follow:
generator 1
-1 * (1,6)
1 * (1,4)
...
generator 2
-1 * (1,6)
1 * (1,4)
...

where each line contains a coefficient and a simplex. Each line is converted, using regular expressions, from a string to a pair (coefficient, Simplex), like

(-1, (1,6))

representing an element in the free abelian group with basis given by simplices. Each generator is a list of such pairs, representing the sum of such elements. These are reassembled in CHomP.__call__() to actual elements in the free module generated by the simplices of the simplicial complex in the appropriate dimension.

Therefore the return value is a list of lists of pairs, one list of pairs for each generator.

EXAMPLES:

sage: from sage.interfaces.chomp import process_generators_simplicial
sage: s = "The 2 generators of H_1 follow:\ngenerator 1:\n-1 * (1,6)\n1 * (1,4)"
sage: process_generators_simplicial(s, 1, simplicial_complexes.Torus())
[[(-1, (1, 6)), (1, (1, 4))]]