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#clang C++
#clib braiding
r"""
Cython wrapper for the libbraiding library.
The libbraiding library is a modification of the braiding program
by Juan Gonzalez-Meneses (https://github.com/jeanluct/cbraid)
to expose the functions as a C++ library instead of an interactive
program.
Braids are returned in left normal form as a list of lists. The
first list contains only an integer, representing the power of
Delta. The subsequent lists are the Tietze lists of the elementary
permutation braids.
"""
from libcpp.list cimport list
cdef extern from "braiding.h" namespace "Braiding":
list[list[int]] ConjugatingBraid (int n, list[int] word, list[int] word2)
list[list[int]] LeftNormalForm (int n, list[int] word)
list[list[int]] RightNormalForm (int n, list[int] word)
list[list[int]] GreatestCommonDivisor(int n, list[int] word1, list[int] word2)
list[list[int]] LeastCommonMultiple(int n, list[int] word1, list[int] word2)
list[list[list[int]]] CentralizerGenerators(int n, list[int] word)
list[list[list[int]]] SuperSummitSet(int n, list[int] word)
list[list[list[list[int]]]] UltraSummitSet(int n, list[int] word)
int thurstontype(int n, list[int] word);
int Rigidity_ext(int n, list[int] word);
list[list[list[list[int]]]] SlidingCircuits(int n, list[int] word)
def conjugatingbraid(braid1, braid2):
r"""
Return a braid that conjugates braid1 to braid2, if such a braid exists.
INPUT:
- ``braid1`` -- the braid to be conjugated.
- ``braid2`` -- the braid to conjugate to.
OUTPUT:
The list of lists that represent a conjugating braid. If the input braids
are not conjugate, an empty list is returned.
EXAMPLES::
sage: B = BraidGroup(3)
sage: b = B([1,2,1,-2])
sage: c = B([1,2])
sage: conjugatingbraid(b,c) # optional - libbraiding
[[0], [2]]
"""
nstrands = max(braid1.parent().strands(), braid2.parent().strands())
l1 = braid1.Tietze()
l2 = braid2.Tietze()
sig_on()
cdef list[list[int]] rop = ConjugatingBraid(nstrands, l1, l2)
sig_off()
return rop
def leftnormalform(braid):
r"""
Return the left normal form of a braid.
INPUT:
- ``braid`` -- a braid
OUTPUT:
A list of lists with the left normal form. The first list contains the
power of delta. The subsequent lists are the elementary permutation braids.
EXAMPLES::
sage: B = BraidGroup(3)
sage: b = B([1,2,1,-2])
sage: leftnormalform(b) # optional - libbraiding
[[0], [2, 1]]
"""
nstrands = braid.parent().strands()
l1 = braid.Tietze()
sig_on()
cdef list[list[int]] rop = LeftNormalForm(nstrands, l1)
sig_off()
return rop
def rightnormalform(braid):
r"""
Return the right normal form of a braid.
INPUT:
- ``braid`` -- a braid
OUTPUT:
A list of lists with the right normal form. The first list contains the
power of delta. The subsequent lists are the elementary permutation braids.
EXAMPLES::
sage: B = BraidGroup(3)
sage: b = B([1,2,1,-2])
sage: rightnormalform(b) # optional - libbraiding
[[2, 1], [0]]
"""
nstrands = braid.parent().strands()
l1 = braid.Tietze()
sig_on()
cdef list[list[int]] rop = RightNormalForm(nstrands, l1)
sig_off()
return rop
def greatestcommondivisor(braid1, braid2):
r"""
Return the greatest common divisor of two braids.
"""
nstrands = max(braid1.parent().strands(), braid2.parent().strands())
l1 = braid1.Tietze()
l2 = braid2.Tietze()
sig_on()
cdef list[list[int]] rop = GreatestCommonDivisor(nstrands, l1, l2)
sig_off()
return rop
def leastcommonmultiple(braid1, braid2):
r"""
Return the least common multiple of two braids.
"""
nstrands = max(braid1.parent().strands(), braid2.parent().strands())
l1 = braid1.Tietze()
l2 = braid2.Tietze()
sig_on()
cdef list[list[int]] rop = LeastCommonMultiple(nstrands, l1, l2)
sig_off()
return rop
def centralizer(braid):
r"""
Return a list of generators of the centralizer of a braid.
"""
nstrands = braid.parent().strands()
lnf = leftnormalform(braid)
if len(lnf) == 1: # (lib)braiding crashes when the input is a power of Delta.
if lnf[0][0] % 2 == 0:
return [[[0], [i+1]] for i in range(nstrands)]
elif nstrands % 2:
return [[[0], [i+1, nstrands - i -1]] for i in range(nstrands/2)]
else:
return [[[0], [i+1, nstrands - i -1]] for i in range(nstrands/2-1)] + [[[0], [nstrands/2]]]
l = braid.Tietze()
sig_on()
cdef list[list[list[int]]] rop = CentralizerGenerators(nstrands, l)
sig_off()
return rop
def supersummitset(braid):
r"""
Return a list with the super-summit-set of a braid.
"""
nstrands = braid.parent().strands()
l = braid.Tietze()
sig_on()
cdef list[list[list[int]]] rop = SuperSummitSet(nstrands, l)
sig_off()
return rop
def ultrasummitset(braid):
r"""
Return a list with the ultra-summit-set of the braid.
"""
nstrands = braid.parent().strands()
l = braid.Tietze()
sig_on()
cdef list[list[list[list[int]]]] rop = UltraSummitSet(nstrands, l)
sig_off()
return rop
def thurston_type(braid):
r"""
Return the Thurston type of the braid
"""
nstrands = braid.parent().strands()
l = braid.Tietze()
sig_on()
cdef int i = thurstontype(nstrands, l)
sig_off()
if i == 1:
return 'periodic'
elif i==2:
return 'reducible'
elif i==3:
return 'pseudo-anosov'
def rigidity(braid):
r"""
Return the rigidity of the braid
"""
nstrands = braid.parent().strands()
l = braid.Tietze()
sig_on()
cdef int i = Rigidity_ext(nstrands, l)
sig_off()
return i
def sliding_circuits(braid):
r"""
Return the set of sliding circuits of the braid
"""
nstrands = braid.parent().strands()
l = braid.Tietze()
sig_on()
cdef list[list[list[list[int]]]] rop = SlidingCircuits(nstrands, l)
sig_off()
return rop
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