def __init__(self, names):
"""
Python constructor.
INPUT:
- ``names`` -- a tuple of strings. The names of the
generators.
TESTS::
sage: B1 = BraidGroup(5) # indirect doctest
sage: B1
Braid group on 5 strands
Check that :trac:`14081` is fixed::
sage: BraidGroup(2)
Braid group on 2 strands
sage: BraidGroup(('a',))
Braid group on 2 strands
"""
n = len(names)
if n<1: #n is the number of generators, not the number of strands (see ticket 14081)
raise ValueError("the number of strands must be an integer bigger than one")
free_group = FreeGroup(names)
rels = []
for i in range(1, n):
if i<n-1:
rels.append(free_group([i, i+1, i, -i-1, -i, -i-1]))
for j in range(i+2, n):
rels.append(free_group([i, j, -i, -j]))
FinitelyPresentedGroup.__init__(self, free_group, tuple(rels))
self._nstrands_ = n+1
def __init__(self, coxeter_matrix, names):
"""
Initialize ``self``.
TESTS::
sage: A = ArtinGroup(['D',4])
sage: TestSuite(A).run()
sage: A = ArtinGroup(['B',3], ['x','y','z'])
sage: TestSuite(A).run()
"""
self._coxeter_group = CoxeterGroup(coxeter_matrix)
free_group = FreeGroup(names)
rels = []
# Generate the relations based on the Coxeter graph
I = coxeter_matrix.index_set()
for ii, i in enumerate(I):
for j in I[ii + 1:]:
m = coxeter_matrix[i, j]
if m == Infinity: # no relation
continue
elt = [i, j] * m
for ind in range(m, 2 * m):
elt[ind] = -elt[ind]
rels.append(free_group(elt))
FinitelyPresentedGroup.__init__(self, free_group, tuple(rels))
def __init__(self, names):
"""
Python constructor.
INPUT:
- ``names`` -- a tuple of strings. The names of the
generators.
TESTS::
sage: B1 = BraidGroup(5) # indirect doctest
sage: B1
Braid group on 5 strands
"""
n = len(names)
if n<2:
raise ValueError("n must be an integer bigger than one")
free_group = FreeGroup(names)
rels = []
for i in range(1, n):
if i<n-1:
rels.append(free_group([i, i+1, i, -i-1, -i, -i-1]))
for j in range(i+2, n):
rels.append(free_group([i, j, -i, -j]))
FinitelyPresentedGroup.__init__(self, free_group, tuple(rels))
self._nstrands_ = n+1
def __init__(self, coxeter_matrix, names):
"""
Initialize ``self``.
TESTS::
sage: A = ArtinGroup(['D',4])
sage: TestSuite(A).run()
sage: A = ArtinGroup(['B',3], ['x','y','z'])
sage: TestSuite(A).run()
"""
self._coxeter_group = CoxeterGroup(coxeter_matrix)
free_group = FreeGroup(names)
rels = []
# Generate the relations based on the Coxeter graph
I = coxeter_matrix.index_set()
for ii,i in enumerate(I):
for j in I[ii+1:]:
m = coxeter_matrix[i,j]
if m == Infinity: # no relation
continue
elt = [i,j]*m
for ind in range(m, 2*m):
elt[ind] = -elt[ind]
rels.append(free_group(elt))
FinitelyPresentedGroup.__init__(self, free_group, tuple(rels))
def __init__(self, G, names):
"""
Initialize ``self``.
TESTS::
sage: G = RightAngledArtinGroup(graphs.CycleGraph(5))
sage: TestSuite(G).run()
"""
self._graph = G
F = FreeGroup(names=names)
CG = Graph(G).complement() # Make sure it's mutable
CG.relabel() # Standardize the labels
cm = [[-1] * CG.num_verts() for _ in range(CG.num_verts())]
for i in range(CG.num_verts()):
cm[i][i] = 1
for u, v in CG.edge_iterator(labels=False):
cm[u][v] = 2
cm[v][u] = 2
self._coxeter_group = CoxeterGroup(
CoxeterMatrix(cm, index_set=G.vertices()))
rels = tuple(
F([i + 1, j + 1, -i - 1, -j - 1])
for i, j in CG.edge_iterator(labels=False)) # +/- 1 for indexing
FinitelyPresentedGroup.__init__(self, F, rels)
def __init__(self, names):
"""
Python constructor.
INPUT:
- ``names`` -- a tuple of strings. The names of the
generators.
TESTS::
sage: B1 = BraidGroup(5) # indirect doctest
sage: B1
Braid group on 5 strands
sage: TestSuite(B1).run()
Check that :trac:`14081` is fixed::
sage: BraidGroup(2)
Braid group on 2 strands
sage: BraidGroup(('a',))
Braid group on 2 strands
Check that :trac:`15505` is fixed::
sage: B=BraidGroup(4)
sage: B.relations()
(s0*s1*s0*s1^-1*s0^-1*s1^-1, s0*s2*s0^-1*s2^-1, s1*s2*s1*s2^-1*s1^-1*s2^-1)
sage: B=BraidGroup('a,b,c,d,e,f')
sage: B.relations()
(a*b*a*b^-1*a^-1*b^-1,
a*c*a^-1*c^-1,
a*d*a^-1*d^-1,
a*e*a^-1*e^-1,
a*f*a^-1*f^-1,
b*c*b*c^-1*b^-1*c^-1,
b*d*b^-1*d^-1,
b*e*b^-1*e^-1,
b*f*b^-1*f^-1,
c*d*c*d^-1*c^-1*d^-1,
c*e*c^-1*e^-1,
c*f*c^-1*f^-1,
d*e*d*e^-1*d^-1*e^-1,
d*f*d^-1*f^-1,
e*f*e*f^-1*e^-1*f^-1)
"""
n = len(names)
if n < 1: #n is the number of generators, not the number of strands (see ticket 14081)
raise ValueError(
"the number of strands must be an integer bigger than one")
free_group = FreeGroup(names)
rels = []
for i in range(1, n):
rels.append(free_group([i, i + 1, i, -i - 1, -i, -i - 1]))
for j in range(i + 2, n + 1):
rels.append(free_group([i, j, -i, -j]))
FinitelyPresentedGroup.__init__(self, free_group, tuple(rels))
self._nstrands_ = n + 1
def __init__(self, names):
"""
Python constructor.
INPUT:
- ``names`` -- a tuple of strings. The names of the
generators.
TESTS::
sage: B1 = BraidGroup(5) # indirect doctest
sage: B1
Braid group on 5 strands
sage: TestSuite(B1).run()
Check that :trac:`14081` is fixed::
sage: BraidGroup(2)
Braid group on 2 strands
sage: BraidGroup(('a',))
Braid group on 2 strands
Check that :trac:`15505` is fixed::
sage: B=BraidGroup(4)
sage: B.relations()
(s0*s1*s0*s1^-1*s0^-1*s1^-1, s0*s2*s0^-1*s2^-1, s1*s2*s1*s2^-1*s1^-1*s2^-1)
sage: B=BraidGroup('a,b,c,d,e,f')
sage: B.relations()
(a*b*a*b^-1*a^-1*b^-1,
a*c*a^-1*c^-1,
a*d*a^-1*d^-1,
a*e*a^-1*e^-1,
a*f*a^-1*f^-1,
b*c*b*c^-1*b^-1*c^-1,
b*d*b^-1*d^-1,
b*e*b^-1*e^-1,
b*f*b^-1*f^-1,
c*d*c*d^-1*c^-1*d^-1,
c*e*c^-1*e^-1,
c*f*c^-1*f^-1,
d*e*d*e^-1*d^-1*e^-1,
d*f*d^-1*f^-1,
e*f*e*f^-1*e^-1*f^-1)
"""
n = len(names)
if n<1: #n is the number of generators, not the number of strands (see ticket 14081)
raise ValueError("the number of strands must be an integer bigger than one")
free_group = FreeGroup(names)
rels = []
for i in range(1, n):
rels.append(free_group([i, i+1, i, -i-1, -i, -i-1]))
for j in range(i+2, n+1):
rels.append(free_group([i, j, -i, -j]))
FinitelyPresentedGroup.__init__(self, free_group, tuple(rels))
self._nstrands_ = n+1
def __init__(self, G):
"""
Initialize ``self``.
INPUT:
- ``G`` -- a graph
TESTS::
sage: G = RightAngledArtinGroup(graphs.CycleGraph(5))
sage: TestSuite(G).run()
"""
self._graph = G
F = FreeGroup(names=['v{}'.format(v) for v in self._graph.vertices()])
CG = Graph(G).complement() # Make sure it's mutable
CG.relabel() # Standardize the labels
rels = tuple(F([i+1, j+1, -i-1, -j-1]) for i,j in CG.edges(False)) #+/- 1 for indexing
FinitelyPresentedGroup.__init__(self, F, rels)
def __init__(self, G):
"""
Initialize ``self``.
INPUT:
- ``G`` -- a graph
TESTS::
sage: G = RightAngledArtinGroup(graphs.CycleGraph(5))
sage: TestSuite(G).run()
"""
self._graph = G
F = FreeGroup(names=['v{}'.format(v) for v in self._graph.vertices()])
CG = Graph(G).complement() # Make sure it's mutable
CG.relabel() # Standardize the labels
rels = tuple(F([i+1, j+1, -i-1, -j-1]) for i,j in CG.edges(False)) #+/- 1 for indexing
FinitelyPresentedGroup.__init__(self, F, rels)
def __init__(self, G, names):
"""
Initialize ``self``.
TESTS::
sage: G = RightAngledArtinGroup(graphs.CycleGraph(5))
sage: TestSuite(G).run()
"""
self._graph = G
F = FreeGroup(names=names)
CG = Graph(G).complement() # Make sure it's mutable
CG.relabel() # Standardize the labels
cm = [[-1]*CG.num_verts() for _ in range(CG.num_verts())]
for i in range(CG.num_verts()):
cm[i][i] = 1
for u,v in CG.edge_iterator(labels=False):
cm[u][v] = 2
cm[v][u] = 2
self._coxeter_group = CoxeterGroup(CoxeterMatrix(cm, index_set=G.vertices()))
rels = tuple(F([i + 1, j + 1, -i - 1, -j - 1])
for i, j in CG.edge_iterator(labels=False)) # +/- 1 for indexing
FinitelyPresentedGroup.__init__(self, F, rels)
