def example_3_1b():
''' Nathans origional version. '''
T = flipper.create_triangulation([[1, 2, 5], [0, 6, 3], [4, ~1, 7],
[~3, 8, ~2], [9, ~5, ~6], [10, ~0, ~9],
[~10, ~7, ~8], [11, 12, ~4],
[~12, 14, 13], [~13, ~11, ~14]])
a = T.lamination([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1])
b = T.lamination([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1])
c = T.lamination([0, 1, 1, 0, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1])
d = T.lamination([0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0])
e = T.lamination([1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0])
f = T.lamination([0, 1, 1, 1, 2, 0, 1, 1, 0, 1, 1, 2, 2, 2, 2])
g = T.lamination([0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 0, 0, 0, 0])
h = T.lamination([0, 1, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0])
return flipper.kernel.EquippedTriangulation(T, [a, b, c, d, e, f, g, h], [
a.encode_twist(),
b.encode_twist(),
c.encode_twist(),
d.encode_twist(),
e.encode_twist(),
f.encode_twist(),
g.encode_twist(),
h.encode_twist()
])
def example_3_1():
T = flipper.create_triangulation([[0, 6, 1], [7, ~6, ~5], [8, 2, ~7],
[9, ~8, ~4], [10, 3, ~9], [11, ~10, ~3],
[12, 4, ~11], [13, ~12, ~2],
[14, 5, ~13], [~0, ~14, ~1]])
a = T.lamination([0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1])
b = T.lamination([0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0])
c = T.lamination([0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0])
d = T.lamination([0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0])
e = T.lamination([0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0])
f = T.lamination([1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 2, 1, 0])
g = T.lamination([1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2])
h = T.lamination([1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 2])
return flipper.kernel.EquippedTriangulation(T, [a, b, c, d, e, f, g, h], [
a.encode_twist(),
b.encode_twist(),
c.encode_twist(),
d.encode_twist(),
e.encode_twist(),
f.encode_twist(),
g.encode_twist(),
h.encode_twist()
])
def example_24():
# A 24-gon.
T = flipper.create_triangulation([[12, 13, 0], [14, 1, ~13], [15, 2, ~14],
[~15, 16, 3], [17, 4, ~16], [~17, 18, 5],
[~18, 19, 6], [20, 7, ~19], [21, 8, ~20],
[~21, 22, 9], [~22, 23, 10],
[24, 11, ~23], [25, ~0, ~24],
[~25, 26, ~1], [~26, 27, ~2],
[28, ~3, ~27], [29, ~4, ~28],
[~29, 30, ~5], [~30, 31, ~6],
[32, ~7, ~31], [33, ~8, ~32],
[~33, 34, ~9], [~34, 35, ~10],
[~12, ~11, ~35]])
a = T.lamination([
0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
])
b = T.lamination([
0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
])
p = T.find_isometry(T, {13: ~12}) # This is a 1/24 click.
return flipper.kernel.EquippedTriangulation(T, [a, b], {
'a': a.encode_twist(),
'b': b.encode_twist(),
'p': p.encode()
})
def example_1_1():
T = flipper.create_triangulation([[0, 2, 1], [~0, ~2, ~1]])
a = T.lamination([1, 0, 1])
b = T.lamination([0, 1, 1])
return flipper.kernel.EquippedTriangulation(
T, [a, b], [a.encode_twist(), b.encode_twist()])
def example_1_1m():
# Mirror image of S_1_1 and its standard (Twister) curves:
T = flipper.create_triangulation([[0, 1, 2], [~0, ~1, ~2]])
a = T.lamination([1, 1, 0])
b = T.lamination([0, 1, 1])
return flipper.kernel.EquippedTriangulation(
T, [a, b], [a.encode_twist(), b.encode_twist()])
def example_1_2p():
# S_1_2 without the half twist
T = flipper.create_triangulation([[1, 3, 2], [~2, 0, 4], [~1, 5, ~0],
[~5, ~4, ~3]])
a = T.lamination([0, 0, 1, 1, 1, 0])
b = T.lamination([1, 0, 0, 0, 1, 1])
c = T.lamination([0, 1, 0, 1, 0, 1])
return flipper.kernel.EquippedTriangulation(
T, [a, b, c], [a.encode_twist(),
b.encode_twist(),
c.encode_twist()])
def example_braid_sphere(n):
# A triangulation of S_{0,n}.
assert (isinstance(n, flipper.IntegerType))
assert (n >= 4)
# We'll build the following triangulation of S_{0,n}:
#
# | | | | |
# |0 |1 |2 |3 |n-3
# | | | | |
# | 2n-4 | 2n-3 | 2n-2 | | 3n-7
# X------X------X------X- ... -X------X
# | | | | |
# | | | | |
# |n-2 |n-1 |n |2n-6 |2n-5
# | | | | |
#
# Note that there puncture 1 is not shown here and is on the "boundary" of the disk.
# There are two families of triangles on the top and the bottom and two exceptional cases:
# 1) Top triangles are given by [i, i+2n-4, ~(i+1)] for i in range(0, n-3)],
# 2) Bottom triangles are given by [i, ~(i+1), ~(i+n-1)] for i in range(n-2, 2n-5),
# 3) Left triangle given by [~0, ~(n-2), ~(2n-4)], and
# 4) Right triangle given by [n-3, 3n-7, 2n-5].
T = flipper.create_triangulation(
[[i, i + 2 * n - 4, ~(i + 1)]
for i in range(0, n - 3)] + [[i, ~(i + 1), ~(i + n - 1)]
for i in range(n - 2, 2 * n - 5)] +
[[~0, ~(n - 2), ~(2 * n - 4)], [n - 3, 3 * n - 7, 2 * n - 5]])
# We'll then create a curve isolating the ith and (i+1)st punctures from the others.
# There are four special cases and a generic case:
# 0) A curve isolating punctures 1 & 2, meeting edges 1, 2, ..., n-3, n-2, n-1, ..., 2n-5, 2n-4,
# 1) A curve isolating punctures 2 & 3, meeting edges 0, 1, n-2 and 2n-3,
# 2) A curve isolating punctures i & i+1, meeting edges i, i+1, i+n-3, i+n-2, i+2n-5 and i+2n-3,
# 3) A curve isolating punctures n-1 & n, meeting edges n-3, 2n-6, 2n-5 and 3n-7, and
# 4) A curve isolating punctures n & 1, meeting edges 0, 1, ..., n-3, n-2, n-1, ..., 2n-6, 3n-7.
laminations = \
[T.lamination([1 if 1 <= j <= 2 * n - 4 else 0 for j in range(T.zeta)])] + \
[T.lamination([1 if j in [0, 1, n-2, 2*n - 3] else 0 for j in range(T.zeta)])] + \
[T.lamination([1 if j in [i, i+1, i+n-3, i+n-2, i+2*n-5, i+2*n-3] else 0 for j in range(T.zeta)]) for i in range(1, n-3)] + \
[T.lamination([1 if j in [n-3, 2*n - 6, 2*n - 5, 3*n - 8] else 0 for j in range(T.zeta)])] + \
[T.lamination([1 if 0 <= j <= 2 * n - 6 or j == 3*n - 7 else 0 for j in range(T.zeta)])]
# We take the half-twist about each of these curves as the generator \sigma_i of SB_n.
mapping_classes = dict(('s_%d' % index, lamination.encode_halftwist())
for index, lamination in enumerate(laminations))
return flipper.kernel.EquippedTriangulation(T, laminations,
mapping_classes)
def flipper_triangulation(self):
""" Return this surface as a flipper triangulation. """
data = self.triangulation_data()
edges = sorted({d.edge_name for d in data}) # Remove duplicates.
edge_lookup = dict((edge, index) for index, edge in enumerate(edges))
ans = [
tuple(edge_lookup[datum.edge_name] if datum.sign ==
+1 else ~edge_lookup[datum.edge_name] for datum in x)
for x in zip(data[::3], data[1::3], data[2::3])
]
return flipper.create_triangulation(ans)
def example_0_4():
T = flipper.create_triangulation([[0, 3, ~0], [1, 4, ~3], [~1, ~4, 5],
[~2, ~5, 2]])
a = T.lamination([0, 1, 0, 0, 1, 0])
b = T.lamination([1, 0, 1, 2, 2, 2])
return flipper.kernel.EquippedTriangulation(
T, [a, b], {
'a': a.encode_twist(),
'b': b.encode_twist(),
'w': a.encode_halftwist(),
'x': b.encode_halftwist(),
'y': a.encode_halftwist(),
'z': b.encode_halftwist()
})
def example_12():
# A 12-gon:
T = flipper.create_triangulation([[6, 7, 0], [8, 1, ~7], [~8, 9, 2],
[~9, 10, 3], [11, 4, ~10], [12, 5, ~11],
[~12, 13, ~0], [14, ~1, ~13],
[~14, 15, ~2], [~15, 16, ~3],
[~16, 17, ~4], [~6, ~5, ~17]])
a = T.lamination([1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0])
b = T.lamination([1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0])
p = T.find_isometry(T, {7: ~6}) # This is a 1/12 click.
return flipper.kernel.EquippedTriangulation(T, [a, b], {
'a': a.encode_twist(),
'b': b.encode_twist(),
'p': p.encode()
})
def example_4_1():
T = flipper.create_triangulation([[~8, 1, 2], [3, 4, ~9], [6, ~10, 5],
[~0, ~11, 7], [~12, ~1,
~2], [~13, ~3, ~4],
[~14, ~5, ~6], [~15, ~7, 0], [8, 9, ~16],
[11, ~17, 10], [13, ~18, 12],
[~19, 14, 15], [~20, 16, 17],
[18, 19, 20]])
a = T.lamination(
[0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2])
b = T.lamination(
[0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2])
c = T.lamination(
[0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2])
d = T.lamination(
[0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2])
e = T.lamination(
[0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2])
f = T.lamination(
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1])
g = T.lamination(
[0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1])
h = T.lamination(
[1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 2, 1, 0, 0, 1, 2, 1, 0, 1])
i = T.lamination(
[1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 0, 0, 1, 2, 2, 1, 0, 1, 2, 1])
j = T.lamination(
[1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 2])
return flipper.kernel.EquippedTriangulation(T,
[a, b, c, d, e, f, g, h, i, j],
[
a.encode_twist(),
b.encode_twist(),
c.encode_twist(),
d.encode_twist(),
e.encode_twist(),
f.encode_twist(),
g.encode_twist(),
h.encode_twist(),
i.encode_twist(),
j.encode_twist()
])
def example_2_1():
# S_2_1 and its standard (Twister) curves:
T = flipper.create_triangulation([[0, 4, 1], [5, ~4, ~3], [2, ~5, 6],
[7, ~6, ~2], [3, ~7, 8], [~0, ~8, ~1]])
a = T.lamination([0, 1, 1, 0, 1, 1, 2, 1, 1])
b = T.lamination([0, 1, 0, 1, 1, 0, 0, 0, 1])
c = T.lamination([1, 0, 0, 1, 1, 0, 0, 0, 1])
d = T.lamination([0, 1, 1, 1, 1, 0, 1, 2, 1])
e = T.lamination([0, 1, 1, 1, 1, 2, 1, 0, 1])
f = T.lamination([0, 0, 1, 0, 0, 0, 1, 0, 0])
return flipper.kernel.EquippedTriangulation(T, [a, b, c, d, e, f], [
a.encode_twist(),
b.encode_twist(),
c.encode_twist(),
d.encode_twist(),
e.encode_twist(),
f.encode_twist()
])
def example_2_1b():
''' Nathans origional version. '''
T = flipper.create_triangulation([[1, 2, 4], [5, 3, 0], [~2, 6, ~1],
[~3, ~0, 7], [~4, ~5, 8], [~7, ~8, ~6]])
a = T.lamination([0, 1, 1, 0, 0, 0, 0, 0, 0])
b = T.lamination([0, 0, 1, 0, 1, 0, 1, 0, 1])
c = T.lamination([1, 0, 0, 0, 0, 1, 0, 1, 1])
d = T.lamination([0, 1, 1, 1, 0, 1, 2, 1, 1])
e = T.lamination([0, 1, 1, 1, 2, 1, 0, 1, 1])
f = T.lamination([0, 1, 2, 1, 1, 1, 1, 1, 0])
return flipper.kernel.EquippedTriangulation(T, [a, b, c, d, e, f], [
a.encode_twist(),
b.encode_twist(),
c.encode_twist(),
d.encode_twist(),
e.encode_twist(),
f.encode_twist()
])
def example_36():
# A 36-gon
T = flipper.create_triangulation([[18, 19, 0], [20, 1, ~19], [21, 2, ~20],
[~21, 22, 3], [~22, 23, 4], [24, 5, ~23],
[25, 6, ~24], [~25, 26, 7], [27, 8, ~26],
[~27, 28, 9], [~28, 29, 10],
[30, 11, ~29], [31, 12, ~30],
[~31, 32, 13], [~32, 33, 14],
[34, 15, ~33], [35, 16, ~34],
[~35, 36, 17], [~36, 37, ~0],
[38, ~1, ~37], [39, ~2, ~38],
[~39, 40, ~3], [~40, 41, ~4],
[42, ~5, ~41], [43, ~6, ~42],
[~43, 44, ~7], [~44, 45, ~8],
[46, ~9, ~45], [47, ~10, ~46],
[~47, 48, ~11], [~48, 49, ~12],
[50, ~13, ~49], [51, ~14, ~50],
[~51, 52, ~15], [~52, 53, ~16],
[~18, ~17, ~53]])
a = T.lamination([
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0
])
b = T.lamination([
0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0
])
p = T.find_isometry(T, {19: ~18}) # This is a 1/36 click.
return flipper.kernel.EquippedTriangulation(T, [a, b], {
'a': a.encode_twist(),
'b': b.encode_twist(),
'p': p.encode()
})
def example_1_2():
# S_1_2 and its standard (Twister) curves:
T = flipper.create_triangulation([[1, 3, 2], [~2, 0, 4], [~1, 5, ~0],
[~5, ~4, ~3]])
a = T.lamination([0, 0, 1, 1, 1, 0])
b = T.lamination([1, 0, 0, 0, 1, 1])
c = T.lamination([0, 1, 0, 1, 0, 1])
x = T.lamination([2, 0, 2, 2, 2, 2])
return flipper.kernel.EquippedTriangulation(
T, {
'a': a,
'b': b,
'c': c,
'x': x
}, {
'a': a.encode_twist(),
'b': b.encode_twist(),
'c': c.encode_twist(),
'x': T.encode([{
0: ~2
}, 4, 5, 2, 3, 0, 4, 2])
})
def LP(genus, num_punctures):
g = int(genus)
n = int(num_punctures)
# The following functions make edge labelling easier.
# In the picture of the triangulation in documentation, some edges are labelled with colored numbers.
# The below function convert these coloured numbers to indices.
# For example, if an edge of the triangulation is labelled 5 and colored green then its index is 5+2*g (or 2 if g = 1).
green = lambda m: m + 2 * g if g > 1 else 2
pink = lambda m: m + 4 * g if g > 2 else 8
blue = lambda m: m + 5 * g
orange = lambda m: m + 6 * g - 3
empty_lam = lambda: [0 for i in range(6 * g + 3 * n - 6)]
if g == 0:
triangles = [(3 * i, 3 * i + 2, ~(3 * ((i + 1) % (n - 2))))
for i in range(n - 2)] + [(3 * i + 1, ~(3 * (
(i + 1) % (n - 2)) + 1), ~(3 * i + 2))
for i in range(n - 2)]
lams = dict()
for i in range(n):
lams['q' + str(i)] = empty_lam()
for j in range(3, 3 * n - 8, 3) + [1, 2, 3 * n - 7]:
lams['q0'][j] += 1
for j in range(3, 6) + [0, 1, 3 * n - 7]:
lams['q1'][j] += 1
for i in range(0, n - 4):
for j in range(3 * i + 2, 3 * i + 5) + range(3 * i + 6, 3 * i + 9):
lams['q' + str(i + 2)][j] += 1
for j in range(1, 3 * n - 8, 3) + [3 * n - 10, 3 * n - 9, 3 * n - 7]:
lams['q' + str(n - 2)][j] += 1
for j in range(1, 3 * n - 7, 3) + range(0, 3 * n - 8,
3) + [3 * n - 7, 3 * n - 7]:
lams['q' + str(n - 1)][j] += 1
else:
if g >= 1:
triangles = [
(k, k + 1, green(k)) for k in range(0, 2 * g - 1, 2)
] + [(~k, ~(k + 1), ~green(k + 1)) for k in range(0, 2 * g - 1, 2)]
if g > 2:
triangles += [(~green(k), green(k + 1), ~pink(k / 2))
for k in range(0, 2 * g - 1, 2)]
if g == 2:
triangles += [(~green(0), green(1), ~pink(0)),
(~green(2), green(3), pink(0))]
if g == 3:
triangles += [(pink(0), pink(1), pink(2))]
if g > 3:
triangles += [
(blue(k), pink(k + 2), ~blue(k + 1)) for k in range(0, g - 4)
] + [(pink(0), pink(1), ~blue(0)),
(pink(g - 2), pink(g - 1), blue(g - 4))]
if n == 2: # For n == 2 we need a certain triangulation, so we can more easily compute the halftwist in terms of flips later.
triangles.remove((2 * g - 2, 2 * g - 1, green(2 * g - 2)))
triangles += [(orange(0), ~orange(2), orange(2)),
(2 * g - 2, ~orange(0), orange(1)),
(green(2 * g - 2), ~orange(1), 2 * g - 1)]
if n > 2:
triangles.remove((2 * g - 2, 2 * g - 1, green(2 * g - 2)))
triangles += [(2 * g - 2, orange(0), orange(1)),
(green(2 * g - 2), ~orange(3 * n - 5),
orange(3 * n - 4)),
(2 * g - 1, ~orange(3 * n - 4), ~orange(3 * n - 6))]
triangles += [(~orange(k), orange(k + 3), ~orange(k + 2))
for k in range(0, 3 * n - 8, 3)
] + [(~orange(k + 1), orange(k + 2), orange(k + 4))
for k in range(0, 3 * n - 8, 3)]
lams = dict()
if g == 1:
lams['a0'] = empty_lam()
lams['a0'][1] += 1
lams['a0'][green(0)] += 1
lams['b0'] = empty_lam()
lams['b0'][0] += 1
lams['b0'][green(0)] += 1
elif g > 1:
for i in [0, 1]:
lams['a' + str(i)] = empty_lam()
for j in [2 * i + 1, green(2 * i), green(2 * i + 1)]:
lams['a' + str(i)][j] += 1
for i in range(g):
lams['b' + str(i)] = empty_lam()
for j in [2 * i, green(2 * i), green(2 * i + 1)]:
lams['b' + str(i)][j] += 1
if g == 2:
lams['c0'] = empty_lam()
for j in [
1,
green(0),
pink(0),
green(2), 2,
green(3),
green(2), 3, 2,
green(2),
pink(0),
green(1)
]:
lams['c0'][j] += 1
elif g > 2:
for i in range(g - 1):
lams['c' + str(i)] = empty_lam()
for j in [
2 * i + 1,
green(2 * i),
pink(i),
pink(i + 1),
green(2 * i + 2), 2 * i + 2,
green(2 * i + 3),
green(2 * i + 2), 2 * i + 3, 2 * i + 2,
green(2 * i + 2),
pink(i + 1),
pink(i),
green(2 * i + 1)
]:
lams['c' + str(i)][j] += 1
for i in range(1, g - 2):
lams['c' + str(i)][blue(i - 1)] += 2
if n == 2:
lams['b' + str(g - 1)][orange(1)] += 1
if 'c' + str(g - 2) in lams.keys():
lams['c' + str(g - 2)][orange(1)] += 2
lams['p0'] = empty_lam()
for j in [
2 * g - 1,
green(2 * g - 1),
green(2 * g - 2),
orange(0),
orange(0),
orange(1),
orange(1),
orange(2)
]:
lams['p0'][j] += 1
if g == 1:
for i in range(n - 1):
lams['p' + str(i)][green(2 * g - 1)] -= 1
lams['q0'] = [2 for i in range(6 * g + 3 * n - 6)]
lams['q0'][orange(0)] -= 2
lams['q0'][orange(2)] -= 2
if n > 2:
if 'a' + str(g - 1) in lams.keys():
lams['a' + str(g - 1)][orange(3 * n - 4)] += 1
for j in [orange(3 * k + 1) for k in range(n - 1)]:
lams['b' + str(g - 1)][j] += 1
if 'c' + str(g - 2) in lams.keys():
for j in [orange(3 * n - 4)] + [
orange(3 * k + 1) for k in range(n - 1)
] + [orange(3 * k + 1) for k in range(n - 1)]:
lams['c' + str(g - 2)][j] += 1
lams['p0'] = empty_lam()
for j in [2 * g - 1, green(2 * g - 1),
green(2 * g - 2)] + [
orange(3 * k) for k in range(n - 1)
] + [orange(3 * k + 1) for k in range(n - 1)]:
lams['p0'][j] += 1
for i in range(1, n - 1):
lams['p' + str(i)] = empty_lam()
for j in [
2 * g - 1,
green(2 * g - 1),
green(2 * g - 2),
orange(3 * i - 1)
] + [orange(3 * k) for k in range(i, n - 1)
] + [orange(3 * k + 1) for k in range(i, n - 1)]:
lams['p' + str(i)][j] += 1
if g == 1:
for i in range(n - 1):
lams['p' + str(i)][green(2 * g - 1)] -= 1
lams['q0'] = [2 for i in range(6 * g + 3 * n - 6)]
for j in [orange(3 * k + 2)
for k in range(1, n - 2)] + [orange(0)]:
lams['q0'][j] -= 2
for j in [orange(3 * k) for k in range(1, n - 1)] + [
orange(3 * k + 1) for k in range(1, n - 1)
] + [orange(2), orange(3 * n - 4)]:
lams['q0'][j] -= 1
lams['q1'] = empty_lam()
for j in [orange(0), orange(1), orange(3), orange(4), orange(5)]:
lams['q1'][j] += 1
for i in range(2, n - 1):
lams['q' + str(i)] = empty_lam()
for j in [
orange(3 * (i - 1) - 1),
orange(3 * (i - 1)),
orange(3 * (i - 1) + 1),
orange(3 * (i - 1) + 3),
orange(3 * (i - 1) + 4),
orange(3 * (i - 1) + 5)
]:
lams['q' + str(i)][j] += 1
triangulation = flipper.create_triangulation(triangles)
# Below we compute the halftwist around the isolating curve bounding the two punctures, for the n == 2 case, in terms of flips.
# We then relable the edges, and re-create the triangulation, since we have altered the original one in the process of finding the flips.
if n == 2:
def _edges_around_vert(t, vert):
""" Return the cyclically ordered list of edges around the given vertex.
Vertex is input as an int corresponding to its index.
That is, for vertex "Puncture 0", do edges_around_vert(t, 0). """
crns = t.corner_class_of_vertex(t.vertices[vert])
edges = [crns[i].edges[1].label for i in range(len(crns))]
edges.reverse()
return edges
t = triangulation
seq = []
vert = 0 if len(_edges_around_vert(t, 0)) > 1 else 1
edges = _edges_around_vert(t, vert)
fixed_edge = orange(2) if orange(2) in edges else ~orange(2)
ind = edges.index(fixed_edge)
edges_0 = [edges[(ind + i) % len(edges)] for i in range(len(edges))]
while len(edges) > 1:
ind = (edges.index(fixed_edge) + 1) % len(edges)
seq.append(edges[ind])
t = t.flip_edge(edges[ind])
edges = _edges_around_vert(t, vert)
edges = _edges_around_vert(t, (vert + 1) % 2)
ind = edges.index(~fixed_edge)
edges = [edges[(ind + i) % len(edges)] for i in range(len(edges))]
isom = dict(zip(edges, edges_0))
seq.append(isom)
seq.reverse()
triangulation = flipper.create_triangulation(triangles)
isolating_halftwist = triangulation.encode(seq)
laminations = dict([(key, triangulation.lamination(value))
for key, value in lams.items()])
twists = dict()
halftwists = dict()
# If n == 2 then we have to add the halftwist (created above from flips) seperately.
# It seems that the halftwist computed above is actually the right Dehn twist (based on the check below using chain relation).
# So it is called 'Q0' and its inverse is 'q0' (lower case indicates left twists, upper case right twists).
if n == 2:
# halftwists['Q0'] = isolating_halftwist
halftwists['q0'] = isolating_halftwist.inverse()
for key in lams.keys():
if key == 'q0' and n == 2:
pass
elif key[0] == 'q':
halftwists[key] = laminations[key].encode_halftwist(1)
# halftwists[key.upper()] = laminations[key].encode_halftwist(-1)
else:
twists[key] = laminations[key].encode_twist(1)
# twists[key.upper()] = laminations[key].encode_twist(-1)
all_twists = dict([(key, value) for key, value in twists.items()] +
[(key, value) for key, value in halftwists.items()])
return flipper.kernel.EquippedTriangulation(triangulation, laminations,
all_twists)
def example_5_1():
T = flipper.create_triangulation([[~10, 1, 2], [~11, 3, 4], [6, ~12, 5],
[7, 8, ~13], [~0, ~14, 9], [~2, ~15, ~1],
[~4, ~16, ~3], [~6, ~17, ~5],
[~8, ~18, ~7], [0, ~19, ~9],
[11, ~20, 10], [12, 13, ~21],
[15, ~22, 14], [~23, 16, 17],
[19, ~24, 18], [20, 21, ~25],
[~26, 22, 23], [24, 25, 26]])
a = T.lamination([
0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0,
1, 1, 0
])
b = T.lamination([
0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0,
1, 1, 0
])
c = T.lamination([
0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0,
1, 2, 1
])
d = T.lamination([
0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1,
1, 2, 1
])
e = T.lamination([
0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1,
1, 2, 1
])
f = T.lamination([
0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1,
1, 2, 1
])
g = T.lamination([
0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2,
0, 2, 2
])
h = T.lamination([
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1,
0, 1, 1
])
i = T.lamination([
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1,
0, 1, 1
])
j = T.lamination([
1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 0, 1, 2, 1,
0, 1, 1
])
k = T.lamination([
1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 0, 0, 1, 2, 2, 2, 1, 0, 1,
2, 1, 1
])
l = T.lamination([
1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0,
2, 2, 0
]) # noqa: E741
return flipper.kernel.EquippedTriangulation(
T, [a, b, c, d, e, f, g, h, i, j, k, l], [
a.encode_twist(),
b.encode_twist(),
c.encode_twist(),
d.encode_twist(),
e.encode_twist(),
f.encode_twist(),
g.encode_twist(),
h.encode_twist(),
i.encode_twist(),
j.encode_twist(),
k.encode_twist(),
l.encode_twist()
])
