Interface to CHomP#
This module is deprecated: see trac ticket #33777.
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
, andhomchain
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 byhomsimpl
andhomcubes
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()
, andhomsimpl()
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 doctest:...: DeprecationWarning: the CHomP interface is deprecated See https://trac.sagemath.org/33777 for details. 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 doctest:...: DeprecationWarning: the CHomP interface is deprecated; hence so is this function See https://trac.sagemath.org/33777 for details. 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
.This function is deprecated: see trac ticket #33777.
- 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 doctest:...: DeprecationWarning: the CHomP interface is deprecated See https://trac.sagemath.org/33777 for details. 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 argumentsubcomplex
is present, compute homology ofcomplex
relative tosubcomplex
.This function is deprecated: see trac ticket #33777.
- 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 doctest:...: DeprecationWarning: the CHomP interface is deprecated See https://trac.sagemath.org/33777 for details. 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 argumentsubcomplex
is present, compute homology ofcomplex
relative tosubcomplex
.This function is deprecated: see trac ticket #33777.
- 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 doctest:...: DeprecationWarning: the CHomP interface is deprecated See https://trac.sagemath.org/33777 for details. 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.
This function is deprecated: see trac ticket #33777.
- 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 elementsai
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 overbase_ring
), withai
representing the unit vector in coordinate \(i-1\). For example, the stringa1 - 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) doctest:...: DeprecationWarning: the CHomP interface is deprecated See https://trac.sagemath.org/33777 for details. [(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.
This function is deprecated: see trac ticket #33777.
- 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 formThe 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) doctest:...: DeprecationWarning: the CHomP interface is deprecated See https://trac.sagemath.org/33777 for details. [[(-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.
This function is deprecated: see trac ticket #33777
- 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 formThe 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()) doctest:...: DeprecationWarning: the CHomP interface is deprecated See https://trac.sagemath.org/33777 for details. [[(-1, (1, 6)), (1, (1, 4))]]