def HNN_splitting(A):
"""
The marked metric graph corresponding to the HNN splitting
F_N=F_{N-1}*<t>.
This is rose marked graph with all edges of length 0 except ``A[0]``
which is of length 1.
INPUT:
- ``A`` alphabet
OUTPUT:
The marked metric graph corresponding to the HNN splitting
EXAMPLES::
sage: A=AlphabetWithInverses(4)
sage: print MarkedMetricGraph.HNN_splitting(A)
Marked graph: a: 0->0, b: 0->0, c: 0->0, d: 0->0
Marking: a->a, b->b, c->c, d->d
Length: a:1, b:0, c:0, d:0
"""
length = dict((a, 0) for a in A.positive_letters())
length[A[0]] = 1
RA = GraphWithInverses.rose_graph(A)
RAA = GraphWithInverses.rose_graph(A)
marking = GraphMap(RA, RAA, dict(
(a, Word([a])) for a in A.positive_letters()))
return MarkedMetricGraph(marking=marking, length=length)
def rose_marked_graph(alphabet):
"""
The rose on ``alphabet`` marked with the identity.
"""
marking=dict((a,a) for a in alphabet.positive_letters())
return MarkedGraph(graph=GraphWithInverses.rose_graph(alphabet),marking=marking,marking_alphabet=alphabet)
def HNN_splitting(A):
"""
The marked metric graph corresponding to the HNN splitting
F_N=F_{N-1}*<t>.
The rose marked graph with all edges of length 0 except ``A[0]``
which is of length 1.
"""
length=dict((a,0) for a in A.positive_letters())
length[A[0]]=1
RA=GraphWithInverses.rose_graph(A)
RAA=GraphWithInverses.rose_graph(A)
marking=GraphMap(RA,RAA,edge_map=dict((a,Word([a])) for a in A.positive_letters()))
return MarkedMetricGraph(marking=marking,length=length)
def rose_map(automorphism):
"""
Returns the core of the rose and the rose acted upon by the corresponding
automorphism.
"""
graph=GraphWithInverses.rose_graph(automorphism._domain._alphabet.copy())
inv_automorphism=automorphism.inverse()
return Core(graph,graph,automorphism,inv_automorphism)
def rose_map(automorphism):
"""
Returns the core of the rose and the rose acted upon by the corresponding
automorphism.
"""
graph = GraphWithInverses.rose_graph(
automorphism._domain._alphabet.copy())
inv_automorphism = automorphism.inverse()
return Core(graph, graph, automorphism, inv_automorphism)
def __init__(self,graph=None,marking=None,alphabet=None,marking_alphabet=None):
if isinstance(marking,GraphMap):
GraphWithInverses.__init__(self,marking.codomain(),marking.codomain().alphabet())
self._marking=marking
else:
if isinstance(graph,GraphWithInverses):
alphabet=graph._alphabet
GraphWithInverses.__init__(self,graph,alphabet)
if marking is None: #computes a (random) marking from a rose equivalent to graph
A=graph.alphabet()
tree=graph.spanning_tree()
j=0
letter=dict()
for a in A.positive_letters():
vi=graph.initial_vertex(a)
vt=graph.terminal_vertex(a)
if (len(tree[vi])==0 or tree[vi][-1]!=A.inverse_letter(a)) and\
(len(tree[vt])==0 or tree[vt][-1]!=a):
letter[j]=a
j=j+1
B=AlphabetWithInverses(j)
RB=GraphWithInverses.rose_graph(B)
edge_map=dict()
for i in xrange(j):
a=letter[i]
edge_map[B[i]]=graph.reduce_path(tree[graph.initial_vertex(a)]\
*Word([a])\
*graph.reverse_path(tree[graph.terminal_vertex(a)]))
marking=GraphMap(RB,graph,edge_map)
else:
marking=GraphMap(GraphWithInverses.rose_graph(marking_alphabet),self,marking)
self._marking=marking
def rose_conjugacy_representative(self):
"""
Topological representative of the conjugacy class of ``self``.
SEE ALSO:
This is the same as ``self.rose_representative()`` but the
base graph of the ``TopologicalRepresentative`` is a
``GraphWithInverses`` instead of a ``MarkedGraph``.
"""
from topological_representative import TopologicalRepresentative
from inverse_graph import GraphWithInverses
return TopologicalRepresentative(GraphWithInverses.rose_graph(self._domain.alphabet()),self)
def rose_conjugacy_representative(self):
"""
Topological representative of the conjugacy class of ``self``.
SEE ALSO:
This is the same as ``self.rose_representative()`` but the
base graph of the ``TopologicalRepresentative`` is a
``GraphWithInverses`` instead of a ``MarkedGraph``.
"""
from topological_representative import TopologicalRepresentative
from inverse_graph import GraphWithInverses
return TopologicalRepresentative(
GraphWithInverses.rose_graph(self._domain.alphabet()), self)
def rose_marked_graph(alphabet):
"""
The rose on ``alphabet`` marked with the identity.
INPUT:
- ``alphabet`` a alphabet
OUTPUT:
The rose on ``alphabet`` marked with the identity.
EXAMPLES::
sage: print MarkedGraph.rose_marked_graph(AlphabetWithInverses(2))
Marked graph: a: 0->0, b: 0->0
Marking: a->a, b->b
"""
marking = dict((a, Word([a])) for a in alphabet.positive_letters())
return MarkedGraph(graph=GraphWithInverses.rose_graph(alphabet),
marking=marking, marking_alphabet=alphabet)
def rose_map(automorphism):
"""
The graph map of the rose representing the automorphism.
The rose is built on a copy of the alphabet of the domain of
``automorphism``.
OUTPUT:
The graph map of the rose representing the automorphism.
EXAMPLES::
sage: phi=FreeGroupAutomorphism('a->ab,b->ac,c->a')
sage: print GraphMap.rose_map(phi)
Graph map:
Graph with inverses: a: 0->0, b: 0->0, c: 0->0
Graph with inverses: a: 0->0, b: 0->0, c: 0->0
edge map: a->ab, b->ac, c->a
"""
graph = GraphWithInverses.rose_graph(
automorphism.domain().alphabet().copy())
return GraphMap(graph, graph, automorphism)
def __init__(self, graph=None, marking=None, alphabet=None,
marking_alphabet=None):
"""
INPUT:
- ``graph`` -- (default None) GraphWithInverses is expected
or will be combute from ''grap''
- ``marking`` -- (default None) GraphMap is expected
or will be compute
- ``alphabet`` -- (default None) if ``graph`` is GraphWithInverses
``alphabet`` will be use for ``self``
- ``marking_alphabet`` -- (default None) alphabet used in the case of a
``MarkedGraph`` is created from a GraphWithInverses`` by
computing (randomly) a rose equivalent to the graph.
EXAMPLES::
sage: G = GraphWithInverses({'a':(0,0),'b':(0,1),'c':(1,0)})
sage: M = MarkedGraph(graph=G)
sage: print M
Marked graph: a: 0->0, c: 1->0, b: 0->1
Marking: a->a, b->bc
sage: A = AlphabetWithInverses(2)
sage: G = GraphWithInverses.rose_graph(A)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->aba,b->ab")
sage: M = MarkedGraph(marking=f)
sage: print M
Marked graph: a: 0->0, b: 0->0
Marking: a->aba, b->ab
sage: print MarkedGraph(marking=f, alphabet=A)
Marked graph: a: 0->0, b: 0->0
Marking: a->aba, b->ab
sage: print MarkedGraph(marking=f, marking_alphabet=A)
Marked graph: a: 0->0, b: 0->0
Marking: a->aba, b->ab
"""
if isinstance(marking, GraphMap):
GraphWithInverses.__init__(self,
marking.codomain(),
marking.codomain().alphabet())
self._marking = marking
else:
if isinstance(graph, GraphWithInverses):
alphabet = graph.alphabet()
GraphWithInverses.__init__(self, graph, alphabet)
if marking is None: # computes a (random) marking
# from a rose equivalent to graph
A = graph.alphabet()
tree = graph.spanning_tree()
j = 0
letter = dict()
for a in A.positive_letters():
vi = graph.initial_vertex(a)
vt = graph.terminal_vertex(a)
if (len(tree[vi]) == 0 or
tree[vi][-1] != A.inverse_letter(a)) \
and (len(tree[vt]) == 0 or tree[vt][-1] != a):
letter[j] = a
j = j + 1
B = AlphabetWithInverses(j)
RB = GraphWithInverses.rose_graph(B)
edge_map = dict()
for i in xrange(j):
a = letter[i]
edge_map[B[i]] = graph.reduce_path(
tree[graph.initial_vertex(a)] * Word([a]) *
graph.reverse_path(tree[graph.terminal_vertex(a)]))
marking = GraphMap(RB, graph, edge_map)
else:
marking = GraphMap(
GraphWithInverses.rose_graph(marking_alphabet),
self, marking)
self._marking = marking
