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r""" 

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 

""" 

from __future__ import print_function 

 

import re 

 

_have_chomp = {} 

def 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 

""" 

global _have_chomp 

if program not in _have_chomp: 

from sage.misc.sage_ostools import have_program 

_have_chomp[program] = have_program(program) 

return _have_chomp[program] 

 

class CHomP: 

""" 

Interface to the CHomP package. 

 

:param program: which CHomP program to use 

:type program: string 

:param complex: a simplicial or cubical complex 

:param subcomplex: a subcomplex of ``complex`` or None (the default) 

:param base_ring: ring over which to perform computations -- must be `\ZZ` or `\GF{p}`. 

:type base_ring: ring; optional, default `\ZZ` 

:param generators: if True, also return list of generators 

:type generators: boolean; optional, default False 

:param verbose: if True, print helpful messages as the computation 

progresses 

:type verbose: boolean; optional, default False 

:param extra_opts: options passed directly to ``program`` 

:type extra_opts: string 

:return: 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 :func:`homchain`, 

:func:`homcubes`, and :func:`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} 

""" 

def __repr__(self): 

""" 

String representation. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.chomp import CHomP 

sage: CHomP() # indirect doctest 

CHomP interface 

""" 

return "CHomP interface" 

 

def __call__(self, program, complex, subcomplex=None, **kwds): 

""" 

Call a CHomP program to compute the homology of a chain 

complex, simplicial complex, or cubical complex. 

 

See :class:`CHomP` for full documentation. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.chomp import CHomP 

sage: T = cubical_complexes.Torus() 

sage: CHomP()('homcubes', T) # indirect doctest, optional - CHomP 

{0: 0, 1: Z x Z, 2: Z} 

""" 

from sage.misc.temporary_file import tmp_filename 

from sage.homology.all import CubicalComplex, cubical_complexes 

from sage.homology.all import SimplicialComplex, Simplex 

from sage.homology.chain_complex import HomologyGroup 

from subprocess import Popen, PIPE 

from sage.rings.all import QQ, ZZ 

from sage.modules.all import VectorSpace, vector 

from sage.combinat.free_module import CombinatorialFreeModule 

 

if not have_chomp(program): 

raise OSError("Program %s not found" % program) 

 

verbose = kwds.get('verbose', False) 

generators = kwds.get('generators', False) 

extra_opts = kwds.get('extra_opts', '') 

base_ring = kwds.get('base_ring', ZZ) 

 

if extra_opts: 

extra_opts = extra_opts.split() 

else: 

extra_opts = [] 

 

# type of complex: 

cubical = False 

simplicial = False 

chain = False 

# CHomP seems to have problems with cubical complexes if the 

# first interval in the first cube defining the complex is 

# degenerate. So replace the complex X with [0,1] x X. 

if isinstance(complex, CubicalComplex): 

cubical = True 

edge = cubical_complexes.Cube(1) 

original_complex = complex 

complex = edge.product(complex) 

if verbose: 

print("Cubical complex") 

elif isinstance(complex, SimplicialComplex): 

simplicial = True 

if verbose: 

print("Simplicial complex") 

else: 

chain = True 

base_ring = kwds.get('base_ring', complex.base_ring()) 

if verbose: 

print("Chain complex over %s" % base_ring) 

 

if base_ring == QQ: 

raise ValueError("CHomP doesn't compute over the rationals, only over Z or F_p.") 

if base_ring.is_prime_field(): 

p = base_ring.characteristic() 

extra_opts.append('-p%s' % p) 

mod_p = True 

else: 

mod_p = False 

 

# 

# complex 

# 

try: 

data = complex._chomp_repr_() 

except AttributeError: 

raise AttributeError("Complex can not be converted to use with CHomP.") 

 

datafile = tmp_filename() 

f = open(datafile, 'w') 

f.write(data) 

f.close() 

 

# 

# subcomplex 

# 

if subcomplex is None: 

if cubical: 

subcomplex = CubicalComplex([complex.n_cells(0)[0]]) 

elif simplicial: 

m = re.search(r'\(([^,]*),', data) 

v = int(m.group(1)) 

subcomplex = SimplicialComplex([[v]]) 

else: 

# replace subcomplex with [0,1] x subcomplex. 

if cubical: 

subcomplex = edge.product(subcomplex) 

# 

# generators 

# 

if generators: 

genfile = tmp_filename() 

extra_opts.append('-g%s' % genfile) 

 

# 

# call program 

# 

if subcomplex is not None: 

try: 

sub = subcomplex._chomp_repr_() 

except AttributeError: 

raise AttributeError("Subcomplex can not be converted to use with CHomP.") 

subfile = tmp_filename() 

f = open(subfile, 'w') 

f.write(sub) 

f.close() 

else: 

subfile = '' 

if verbose: 

print("Popen called with arguments", end="") 

print([program, datafile, subfile] + extra_opts) 

print("") 

print("CHomP output:") 

print("") 

# output = Popen([program, datafile, subfile, extra_opts], 

cmd = [program, datafile] 

if subfile: 

cmd.append(subfile) 

if extra_opts: 

cmd.extend(extra_opts) 

output = Popen(cmd, stdout=PIPE).communicate()[0] 

if verbose: 

print(output) 

print("End of CHomP output") 

print("") 

if generators: 

gens = open(genfile, 'r').read() 

if verbose: 

print("Generators:") 

print(gens) 

# 

# process output 

# 

if output.find('ERROR') != -1: 

raise RuntimeError('error inside CHomP') 

# output contains substrings of one of the forms 

# "H_1 = Z", "H_1 = Z_2 + Z", "H_1 = Z_2 + Z^2", 

# "H_1 = Z + Z_2 + Z" 

if output.find('trivial') != -1: 

if mod_p: 

return {0: VectorSpace(base_ring, 0)} 

else: 

return {0: HomologyGroup(0, ZZ)} 

d = {} 

h = re.compile("^H_([0-9]*) = (.*)$", re.M) 

tors = re.compile("Z_([0-9]*)") 

# 

# homology groups 

# 

for m in h.finditer(output): 

if verbose: 

print(m.groups()) 

# dim is the dimension of the homology group 

dim = int(m.group(1)) 

# hom_str is the right side of the equation "H_n = Z^r + Z_k + ..." 

hom_str = m.group(2) 

# need to read off number of summands and their invariants 

if hom_str.find("0") == 0: 

if mod_p: 

hom = VectorSpace(base_ring, 0) 

else: 

hom = HomologyGroup(0, ZZ) 

else: 

rk = 0 

if hom_str.find("^") != -1: 

rk_srch = re.search(r'\^([0-9]*)\s?', hom_str) 

rk = int(rk_srch.group(1)) 

rk += len(re.findall("(Z$)|(Z\s)", hom_str)) 

if mod_p: 

rk = rk if rk != 0 else 1 

if verbose: 

print("dimension = %s, rank of homology = %s" % (dim, rk)) 

hom = VectorSpace(base_ring, rk) 

else: 

n = rk 

invts = [] 

for t in tors.finditer(hom_str): 

n += 1 

invts.append(int(t.group(1))) 

for i in range(rk): 

invts.append(0) 

if verbose: 

print("dimension = %s, number of factors = %s, invariants = %s" % (dim, n, invts)) 

hom = HomologyGroup(n, ZZ, invts) 

 

# 

# generators 

# 

if generators: 

if cubical: 

g = process_generators_cubical(gens, dim) 

if verbose: 

print("raw generators: %s" % g) 

if g: 

module = CombinatorialFreeModule(base_ring, 

original_complex.n_cells(dim), 

prefix="", 

bracket=True) 

basis = module.basis() 

output = [] 

for x in g: 

v = module(0) 

for term in x: 

v += term[0] * basis[term[1]] 

output.append(v) 

g = output 

elif simplicial: 

g = process_generators_simplicial(gens, dim, complex) 

if verbose: 

print("raw generators: %s" % gens) 

if g: 

module = CombinatorialFreeModule(base_ring, 

complex.n_cells(dim), 

prefix="", 

bracket=False) 

basis = module.basis() 

output = [] 

for x in g: 

v = module(0) 

for term in x: 

if complex._is_numeric(): 

v += term[0] * basis[term[1]] 

else: 

translate = complex._translation_from_numeric() 

simplex = Simplex([translate[a] for a in term[1]]) 

v += term[0] * basis[simplex] 

output.append(v) 

g = output 

elif chain: 

g = process_generators_chain(gens, dim, base_ring) 

if verbose: 

print("raw generators: %s" % gens) 

if g: 

if not mod_p: 

# sort generators to match up with corresponding invariant 

g = [_[1] for _ in sorted(zip(invts, g), key=lambda x: x[0])] 

d[dim] = (hom, g) 

else: 

d[dim] = hom 

else: 

d[dim] = hom 

 

if chain: 

new_d = {} 

diff = complex.differential() 

if len(diff) == 0: 

return {} 

bottom = min(diff) 

top = max(diff) 

for dim in d: 

if complex._degree_of_differential == -1: # chain complex 

new_dim = bottom + dim 

else: # cochain complex 

new_dim = top - dim 

if isinstance(d[dim], tuple): 

# generators included. 

group = d[dim][0] 

gens = d[dim][1] 

new_gens = [] 

dimension = complex.differential(new_dim).ncols() 

# make sure that each vector is embedded in the 

# correct ambient space: pad with a zero if 

# necessary. 

for v in gens: 

v_dict = v.dict() 

if dimension - 1 not in v.dict(): 

v_dict[dimension - 1] = 0 

new_gens.append(vector(base_ring, v_dict)) 

else: 

new_gens.append(v) 

new_d[new_dim] = (group, new_gens) 

else: 

new_d[new_dim] = d[dim] 

d = new_d 

return d 

 

def help(self, program): 

""" 

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

 

:param program: which CHomP program to use 

:type program: string 

:return: 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... 

""" 

from subprocess import Popen, PIPE 

print(Popen([program, '-h'], stdout=PIPE).communicate()[0]) 

 

def homsimpl(complex=None, subcomplex=None, **kwds): 

r""" 

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``. 

 

:param complex: a simplicial complex 

:param subcomplex: a subcomplex of ``complex`` or None (the default) 

:param base_ring: ring over which to perform computations -- must be `\ZZ` or `\GF{p}`. 

:type base_ring: ring; optional, default `\ZZ` 

:param generators: if True, also return list of generators 

:type generators: boolean; optional, default False 

:param verbose: if True, print helpful messages as the computation 

progresses 

:type verbose: boolean; optional, default False 

:param help: if True, just print a help message and exit 

:type help: boolean; optional, default False 

:param extra_opts: options passed directly to ``program`` 

:type extra_opts: string 

:return: 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)])} 

 

TESTS: 

 

Generators for a simplicial complex whose vertices are not integers:: 

 

sage: S1 = simplicial_complexes.Sphere(1) 

sage: homsimpl(S1.join(S1), generators=True, base_ring=GF(2))[3][1] # optional - CHomP 

[('L0', 'L1', 'R0', 'R1') + ('L0', 'L1', 'R0', 'R2') + ('L0', 'L1', 'R1', 'R2') + ('L0', 'L2', 'R0', 'R1') + ('L0', 'L2', 'R0', 'R2') + ('L0', 'L2', 'R1', 'R2') + ('L1', 'L2', 'R0', 'R1') + ('L1', 'L2', 'R0', 'R2') + ('L1', 'L2', 'R1', 'R2')] 

""" 

from sage.homology.all import SimplicialComplex 

help = kwds.get('help', False) 

if help: 

return CHomP().help('homsimpl') 

# Check types here, because if you pass homsimpl a cubical 

# complex, it tries to compute its homology as if it were a 

# simplicial complex and gets terribly wrong answers. 

if (isinstance(complex, SimplicialComplex) 

and (subcomplex is None or isinstance(subcomplex, SimplicialComplex))): 

return CHomP()('homsimpl', complex, subcomplex=subcomplex, **kwds) 

else: 

raise TypeError("Complex and/or subcomplex are not simplicial complexes.") 

 

def homcubes(complex=None, subcomplex=None, **kwds): 

r""" 

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``. 

 

:param complex: a cubical complex 

:param subcomplex: a subcomplex of ``complex`` or None (the default) 

:param base_ring: ring over which to perform computations -- must be `\ZZ` or `\GF{p}`. 

:type base_ring: ring; optional, default `\ZZ` 

:param generators: if True, also return list of generators 

:type generators: boolean; optional, default False 

:param verbose: if True, print helpful messages as the computation progresses 

:type verbose: boolean; optional, default False 

:param help: if True, just print a help message and exit 

:type help: boolean; optional, default False 

:param extra_opts: options passed directly to ``homcubes`` 

:type extra_opts: string 

:return: 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]]] 

""" 

from sage.homology.all import CubicalComplex 

help = kwds.get('help', False) 

if help: 

return CHomP().help('homcubes') 

# Type-checking is here for the same reason as in homsimpl above. 

if (isinstance(complex, CubicalComplex) 

and (subcomplex is None or isinstance(subcomplex, CubicalComplex))): 

return CHomP()('homcubes', complex, subcomplex=subcomplex, **kwds) 

else: 

raise TypeError("Complex and/or subcomplex are not cubical complexes.") 

 

 

def homchain(complex=None, **kwds): 

r""" 

Compute the homology of a chain complex using the CHomP program 

``homchain``. 

 

:param complex: a chain complex 

:param generators: if True, also return list of generators 

:type generators: boolean; optional, default False 

:param verbose: if True, print helpful messages as the computation progresses 

:type verbose: boolean; optional, default False 

:param help: if True, just print a help message and exit 

:type help: boolean; optional, default False 

:param extra_opts: options passed directly to ``homchain`` 

:type extra_opts: string 

:return: 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)]) 

 

TESTS: 

 

Chain complexes concentrated in negative dimensions, cochain complexes, etc.:: 

 

sage: C = ChainComplex({-5: 4 * identity_matrix(ZZ, 2)}, degree=-1) 

sage: homchain(C) # optional - CHomP 

{-6: C4 x C4} 

sage: C = ChainComplex({-5: 4 * identity_matrix(ZZ, 2)}, degree=1) 

sage: homchain(C, generators=True) # optional - CHomP 

{-4: (C4 x C4, [(1, 0), (0, 1)])} 

""" 

from sage.homology.chain_complex import ChainComplex_class 

help = kwds.get('help', False) 

if help: 

return CHomP().help('homchain') 

# Type-checking just in case. 

if isinstance(complex, ChainComplex_class): 

return CHomP()('homchain', complex, **kwds) 

else: 

raise TypeError("Complex is not a chain complex.") 

 

 

def process_generators_cubical(gen_string, dim): 

r""" 

Process CHomP generator information for cubical complexes. 

 

:param gen_string: generator output from CHomP 

:type gen_string: string 

:param dim: dimension in which to find generators 

:type dim: integer 

:return: 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 

:meth:`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 

""" 

from sage.homology.cubical_complex import Cube 

# each dim in gen_string starts with "The generator for 

# H_3 follows:". So search for "T" to find the 

# end of the current list of generators. 

g_srch = re.compile(r'H_%s\sfollow[s]?:\n([^T]*)(?:T|$)' % dim) 

g = g_srch.search(gen_string) 

output = [] 

cubes_list = [] 

if g: 

g = g.group(1) 

if g: 

# project g to one end of the cylinder [0,1] x complex: 

# 

# drop the first coordinate and eliminate duplicates, at least 

# in positive dimensions, drop any line containing a 

# degenerate cube 

g = re.sub('\([01],', '(', g) 

if dim > 0: 

lines = g.splitlines() 

newlines = [] 

for l in lines: 

x = re.findall(r'(\([0-9,]*\))', l) 

if x: 

left, right = x 

left = [int(a) for a in left.strip('()').split(',')] 

right = [int(a) for a in right.strip('()').split(',')] 

if sum([xx - yy for (xx, yy) in zip(right, left)]) == dim: 

newlines.append(l) 

else: # line like "generator 2" 

newlines.append(l) 

g = newlines 

cubes_list = [] 

for l in g: 

cubes = re.search(r'([+-]?)\s?([0-9]+)?\s?[*]?\s?\[\(([-0-9,]+)\)\(([-0-9,]+)\)\]', l) 

if cubes: 

if cubes.group(1) and re.search("-", cubes.group(1)): 

sign = -1 

else: 

sign = 1 

if cubes.group(2) and len(cubes.group(2)) > 0: 

coeff = sign * int(cubes.group(2)) 

else: 

coeff = sign * 1 

cube1 = [int(a) for a in cubes.group(3).split(',')] 

cube2 = [int(a) for a in cubes.group(4).split(',')] 

cube = Cube(zip(cube1, cube2)) 

cubes_list.append((coeff, cube)) 

else: 

if cubes_list: 

output.append(cubes_list) 

cubes_list = [] 

if cubes_list: 

output.append(cubes_list) 

return output 

else: 

return None 

 

def process_generators_simplicial(gen_string, dim, complex): 

r""" 

Process CHomP generator information for simplicial complexes. 

 

:param gen_string: generator output from CHomP 

:type gen_string: string 

:param dim: dimension in which to find generators 

:type dim: integer 

:param complex: simplicial complex under consideration 

:return: 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 

:meth:`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))]] 

""" 

from sage.homology.all import Simplex 

# each dim in gen_string starts with "The generator for H_3 

# follows:". So search for "T" to find the end of the current 

# list of generators. 

g_srch = re.compile(r'H_%s\sfollow[s]?:\n([^T]*)(?:T|$)' % dim) 

g = g_srch.search(gen_string) 

output = [] 

simplex_list = [] 

if g: 

g = g.group(1) 

if g: 

lines = g.splitlines() 

for l in lines: 

simplex = re.search(r'([+-]?)\s?([0-9]+)?\s?[*]?\s?(\([0-9,]*\))', l) 

if simplex: 

if simplex.group(1) and re.search("-", simplex.group(1)): 

sign = -1 

else: 

sign = 1 

if simplex.group(2) and len(simplex.group(2)) > 0: 

coeff = sign * int(simplex.group(2)) 

else: 

coeff = sign * 1 

simplex = Simplex([int(a) for a in simplex.group(3).strip('()').split(',')]) 

simplex_list.append((coeff, simplex)) 

else: 

if simplex_list: 

output.append(simplex_list) 

simplex_list = [] 

if simplex_list: 

output.append(simplex_list) 

return output 

else: 

return None 

 

def process_generators_chain(gen_string, dim, base_ring=None): 

r""" 

Process CHomP generator information for simplicial complexes. 

 

:param gen_string: generator output from CHomP 

:type gen_string: string 

:param dim: dimension in which to find generators 

:type dim: integer 

:param base_ring: base ring over which to do the computations 

:type base_ring: optional, default ZZ 

:return: 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)] 

""" 

from sage.modules.all import vector 

from sage.rings.all import ZZ 

if base_ring is None: 

base_ring = ZZ 

# each dim in gens starts with a string like 

# "[H_3]". So search for "[" to find the end of 

# the current list of generators. 

g_srch = re.compile(r'\[H_%s\]\n([^]]*)(?:\[|$)' % dim) 

g = g_srch.search(gen_string) 

if g: 

g = g.group(1) 

if g: 

# each line in the string g is a linear 

# combination of things like "a2", "a31", etc. 

# indexing on the a's starts at 1. 

lines = g.splitlines() 

new_gens = [] 

for l in lines: 

gen = re.compile(r"([+-]?)\s?([0-9]+)?\s?[*]?\s?a([0-9]*)(?:\s|$)") 

v = {} 

for term in gen.finditer(l): 

if term.group(1) and re.search("-", term.group(1)): 

sign = -1 

else: 

sign = 1 

if term.group(2) and len(term.group(2)) > 0: 

coeff = sign * int(term.group(2)) 

else: 

coeff = sign * 1 

idx = int(term.group(3)) 

v[idx-1] = coeff 

if v: 

new_gens.append(vector(base_ring, v)) 

g = new_gens 

return g