''' Some standard example surfaces with mapping classes defined on them.
Mainly used for running tests on. These can be accessed through
the load(SURFACE) function. '''
import re
import flipper
REGEX_IS_SPHERE_BRAID = re.compile(r'SB_(?P<num_strands>\d+)$')
[docs]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()
})
[docs]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()])
[docs]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()])
[docs]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])
})
[docs]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()])
[docs]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()])
[docs]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()])
[docs]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()])
[docs]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()])
[docs]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()])
[docs]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()])
[docs]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()
})
[docs]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()
})
[docs]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()
})
[docs]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)
[docs]def load(surface):
''' Return the requested example EquippedTriangulation.
Available surfaces:
'S_0_4', 'S_1_1', 'S_1_1m', 'S_1_2', 'S_1_2p',
'S_2_1', 'S_2_1b', 'S_3_1', 'S_3_1b',
'E_12', 'E_24', 'E_36', and
'SB_n' where n is an integer >= 4. '''
surfaces = {
'S_0_4': example_0_4,
'S_1_1': example_1_1,
'S_1_1m': example_1_1m,
'S_1_2': example_1_2,
'S_1_2p': example_1_2p,
'S_2_1': example_2_1,
'S_2_1b': example_2_1b,
'S_3_1': example_3_1,
'S_4_1': example_4_1,
'S_5_1': example_5_1,
'S_3_1b': example_3_1b,
'E_12': example_12,
'E_24': example_24,
'E_36': example_36
}
surface = str(surface)
if surface in surfaces:
return surfaces[surface]()
elif REGEX_IS_SPHERE_BRAID.match(surface):
return example_braid_sphere(int(REGEX_IS_SPHERE_BRAID.match(surface).groupdict()['num_strands']))
raise KeyError('Unknown surface: %s' % surface)
