''' A module for representing and manipulating maps between Triangulations.
Provides one class: Encoding. '''
from itertools import product
import flipper
from flipper.kernel.decorators import memoize # Special import needed for decorating.
NT_TYPE_PERIODIC = 'Periodic'
NT_TYPE_REDUCIBLE = 'Reducible' # Strictly this means "reducible and not periodic".
NT_TYPE_PSEUDO_ANOSOV = 'Pseudo-Anosov'
[docs]class Encoding:
''' This represents a map between two Triagulations.
If it maps to and from the same triangulation then it represents
a mapping class. This can be checked using self.is_mapping_class().
The map is given by a sequence of EdgeFlips, LinearTransformations
and Isometries which act from right to left. '''
def __init__(self, sequence, _cache=None):
assert isinstance(sequence, (list, tuple))
assert sequence
assert all(isinstance(item, flipper.kernel.Move) for item in sequence)
# We used to also test:
# assert all(x.source_triangulation == y.target_triangulation for x, y in zip(sequence, sequence[1:]))
# However this makes composing Encodings a quadratic time algorithm!
self.sequence = sequence
self.source_triangulation = self.sequence[-1].source_triangulation
self.target_triangulation = self.sequence[0].target_triangulation
self.zeta = self.source_triangulation.zeta
self._cache = {'name': ''} if _cache is None else _cache # For caching hard to compute results.
[docs] def without_cache(self):
''' Return this Encoding but with an empty cache. '''
return Encoding(self.sequence, _cache={'name': self._cache['name']})
[docs] def is_mapping_class(self):
''' Return if this encoding is a mapping class.
That is, if it maps to the triangulation it came from. '''
return self.source_triangulation == self.target_triangulation
def __repr__(self):
return str(self)
def __str__(self):
return self._cache['name'] if 'name' in self._cache else str(self.sequence) # A backup name.
def __iter__(self):
return iter(self.sequence)
def __len__(self):
return len(self.sequence)
[docs] def package(self):
''' Return a small amount of info that self.source_triangulation can use to reconstruct this triangulation. '''
return [item.package() for item in self]
def __reduce__(self):
return (create_encoding, (self.source_triangulation, self.package(), self._cache))
[docs] def flip_length(self):
''' Return the number of flips needed to realise this sequence. '''
return sum(item.flip_length for item in self)
def __getitem__(self, value):
if isinstance(value, slice):
# It turns out that handling all slices correctly is really hard.
# We need to be very careful with "empty" slices. As Encodings require
# non-empty sequences, we have to return just the id_encoding. This
# ensures the Encoding that we return satisfies:
# self == self[:i] * self[i:j] * self[j:]
# even when i == j.
start = 0 if value.start is None else value.start if value.start >= 0 else len(self) + value.start
stop = len(self) if value.stop is None else value.stop if value.stop >= 0 else len(self) + value.stop
if start == stop:
if 0 <= start < len(self):
return self.sequence[start].target_triangulation.id_encoding()
else:
raise IndexError('list index out of range')
return Encoding(self.sequence[value])
elif isinstance(value, flipper.IntegerType):
return self.sequence[value]
else:
return NotImplemented
[docs] def identify(self):
''' Return a tuple of integers which uniquely determines this map.
The tuple we return is the intersection numbers of the images of
the key_curves under this map. This uniquely determines the map
(assuming we know source_triangulation and target_triangulation)
by Alexanders trick. '''
if '__identify__' not in self._cache:
images = [self(curve) for curve in self.source_triangulation.key_curves()]
self._cache['__identify__'] = tuple(entry for curve in images for vector in [curve.geometric, curve.algebraic] for entry in vector)
return self._cache['__identify__']
def __eq__(self, other):
if isinstance(other, Encoding):
if self.source_triangulation != other.source_triangulation or self.target_triangulation != other.target_triangulation:
raise ValueError('Cannot compare Encodings between different triangulations.')
return self.identify() == other.identify()
else:
return NotImplemented
def __hash__(self):
return hash(self.identify())
[docs] def is_homologous_to(self, other):
''' Return if this encoding is homologous to other.
Two maps are homologous if and only if they induce the same map
from H_1(source_triangulation) to H_1(target_triangulation). '''
if isinstance(other, Encoding):
if self.source_triangulation != other.source_triangulation or self.target_triangulation != other.target_triangulation:
raise ValueError('Cannot compare Encodings between different triangulations.')
return all(self(curve).is_homologous_to(other(curve)) for curve in self.source_triangulation.key_curves())
else:
return NotImplemented
def __call__(self, other):
if isinstance(other, flipper.kernel.Lamination):
if self.source_triangulation != other.triangulation:
raise ValueError('Cannot apply an Encoding to a Lamination on a triangulation other than source_triangulation.')
geometric = other.geometric
algebraic = other.algebraic
for item in reversed(self.sequence):
geometric = item.apply_geometric(geometric)
algebraic = item.apply_algebraic(algebraic)
return self.target_triangulation.lamination(geometric, algebraic, remove_peripheral=False)
else:
return NotImplemented
def __mul__(self, other):
if isinstance(other, Encoding):
if self.source_triangulation != other.target_triangulation:
raise ValueError('Cannot compose Encodings over different triangulations.')
return Encoding(self.sequence + other.sequence, _cache=dict() if 'name' not in self._cache or 'name' not in other._cache else {'name': self._cache['name'] + '.' + other._cache['name']})
else:
return NotImplemented
def __pow__(self, k):
assert self.is_mapping_class()
if k == 0:
return self.source_triangulation.id_encoding()
elif k > 0:
return Encoding(self.sequence * k, _cache=dict() if 'name' not in self._cache else {'name': '(%s)^%d' % (self._cache['name'], k)})
else:
return self.inverse()**abs(k)
[docs] def inverse(self):
''' Return the inverse of this encoding. '''
return Encoding([item.inverse() for item in reversed(self.sequence)], _cache=dict() if 'name' not in self._cache else {'name': '(%s)^-1' % self._cache['name']})
def __invert__(self):
return self.inverse()
[docs] def closing_isometries(self):
''' Return all the possible isometries from self.target_triangulation to self.source_triangulation.
These are the maps that can be used to close this into a mapping class. '''
return self.target_triangulation.isometries_to(self.source_triangulation)
[docs] def order(self):
''' Return the order of this mapping class.
If this has infinite order then return 0.
This encoding must be a mapping class. '''
assert self.is_mapping_class()
# We could do:
# for i in range(1, self.source_triangulation.max_order + 1):
# if self**i == self.source_triangulation.id_encoding():
# return i
# But this is quadratic in the order so instead we do:
curves = self.source_triangulation.key_curves()
possible_orders = set(range(1, self.source_triangulation.max_order+1))
for curve in curves:
curve_image = curve
for i in range(1, max(possible_orders)+1):
curve_image = self(curve_image)
if curve_image != curve:
possible_orders.discard(i)
if not possible_orders: return 0 # No finite orders remain so we are infinite order.
return min(possible_orders)
[docs] def is_identity(self):
''' Return if this encoding is the identity map. '''
return self.is_mapping_class() and all(self(curve) == curve for curve in self.source_triangulation.key_curves())
[docs] def is_periodic(self):
''' Return if this encoding has finite order.
This encoding must be a mapping class. '''
return self.order() > 0
[docs] def is_reducible(self):
''' Return if this encoding is reducible and NOT periodic.
This encoding must be a mapping class. '''
return self.nielsen_thurston_type() == NT_TYPE_REDUCIBLE
[docs] def is_pseudo_anosov(self):
''' Return if this encoding is pseudo-Anosov.
This encoding must be a mapping class. '''
return self.nielsen_thurston_type() == NT_TYPE_PSEUDO_ANOSOV
[docs] def applied_geometric(self, lamination):
''' Return the action and condition matrices describing the PL map
applied to the geometric coordinates of the given lamination. '''
assert isinstance(lamination, flipper.kernel.Lamination)
As = flipper.kernel.id_matrix(self.zeta)
Cs = flipper.kernel.zero_matrix(self.zeta, 1)
for item in reversed(self.sequence): # We can't just reverse self as reversed requires a list not an iterator.
# This now uses the improved sparse matrix representation.
As, C = item.applied_geometric(lamination, As)
Cs = Cs.join(C)
lamination = item(lamination)
return As, Cs
[docs] def pl_action(self):
''' Yield each of the action, condition matrix pairs describing the action of this Encoding
on ML. '''
for sequence in product(*[range(len(item)) for item in self]):
As = flipper.kernel.id_matrix(self.zeta)
Cs = flipper.kernel.zero_matrix(self.zeta, 1)
for item, index in reversed(list(zip(self, sequence))):
# This now uses the improved sparse matrix representation.
As, C = item.pl_action(index, action=As)
Cs = Cs.join(C)
yield (As, Cs)
[docs] @memoize
def pml_fixedpoint(self):
''' Return a rescaling constant and projectively invariant lamination.
Assumes that the mapping class is pseudo-Anosov.
To find this we start with a curve on the surface and repeatedly apply the map.
We then use :func:`~flipper.kernel.matrix.Matrix.directed_eigenvector()` to find the nearby projective fixed point.
The work of Margalit--Strenner--Yurtas says that if we apply self too many times then self is not pseudo-Anosov.
This encoding must be a mapping class. '''
assert self.is_mapping_class()
# We start with a fast test for periodicity.
# This isn't needed but it means that if we ever discover that self is not pA then it must be reducible.
if self.is_periodic():
raise flipper.AssumptionError('Mapping class is periodic.')
bound = 36 * self.source_triangulation.euler_characteristic**2
for curve in self.source_triangulation.key_curves():
# The result of Margalit--Strenner--Yurtas say that this is a sufficient number of iterations to find a fixed point.
# See https://www.youtube.com/watch?v=-GO0AvUGjH4
for n in range(bound):
curve = self(curve)
if n & (n-1) == 0 or n == bound-1: # if n is a power of two or we have reached the bound.
try:
action_matrix, condition_matrix = self.applied_geometric(curve)
eigenvalue, eigenvector = action_matrix.directed_eigenvector(condition_matrix)
invariant_lamination = self.source_triangulation(eigenvector.tolist())
if invariant_lamination.is_empty(): # But it might have been entirely peripheral.
raise flipper.ComputationError('No interesting eigenvectors in cell.')
return eigenvalue, invariant_lamination
except flipper.ComputationError:
pass
raise flipper.AssumptionError('Mapping class is reducible.')
[docs] def invariant_lamination(self):
''' Return a projectively invariant lamination of this mapping class.
This encoding must be a mapping class.
Assumes that this encoding is pseudo-Anosov. '''
_, lamination = self.pml_fixedpoint()
return lamination
[docs] def dilatation(self):
''' Return the dilatation of this mapping class.
This encoding must be a mapping class. '''
if self.nielsen_thurston_type() != NT_TYPE_PSEUDO_ANOSOV:
return 1
else:
lmbda, _ = self.pml_fixedpoint()
return lmbda
[docs] def splitting_sequences(self, take_roots=False):
''' Return a list of splitting sequences associated to this mapping class.
Assumes (and checks) that the mapping class is pseudo-Anosov.
This encoding must be a mapping class. '''
if self.is_periodic(): # Actually this test is redundant but it is faster to test it now.
raise flipper.AssumptionError('Mapping class is not pseudo-Anosov.')
dilatation, lamination = self.pml_fixedpoint()
try:
splittings = lamination.splitting_sequences(dilatation=None if take_roots else dilatation)
except flipper.AssumptionError as err: # Lamination is not filling.
raise flipper.AssumptionError('Mapping class is not pseudo-Anosov.') from err
return splittings
[docs] def splitting_sequence(self):
''' Return the splitting sequence associated to this mapping class.
Assumes (and checks) that the mapping class is pseudo-Anosov.
This encoding must be a mapping class. '''
# We get a list of all possible splitting sequences from
# self.splitting_sequences(). From there we use the fact that each
# of these differ by a periodic mapping class, which cannot be in the
# Torelli subgroup and so acts non-trivially on H_1(S).
# Thus we look for the map whose action on H_1(S) is conjugate to self
# via splitting.preperiodic.
# To do this we take sufficiently many curves (the key_curves() of the
# underlying triangulation) and look for the splitting sequence in which
# they are mapped to homologous curves by:
# preperiodic . self and periodic . preperiodic.
# Note that we no longer use self.inverse() as periodic now goes in the
# same direction as self.
homology_splittings = [splitting for splitting in self.splitting_sequences() if (splitting.preperiodic * self).is_homologous_to(splitting.mapping_class * splitting.preperiodic)]
if len(homology_splittings) == 0:
raise flipper.FatalError('Mapping class is not homologous to any splitting sequence.')
elif len(homology_splittings) == 1:
return homology_splittings[0]
else: # len(homology_splittings) > 1:
raise flipper.FatalError('Mapping class is homologous to multiple splitting sequences.')
[docs] def canonical(self):
''' Return the canonical form of this mapping class. '''
return self.splitting_sequence().mapping_class
[docs] def nielsen_thurston_type(self):
''' Return the Nielsen--Thurston type of this encoding.
This encoding must be a mapping class. '''
if self.is_periodic():
return NT_TYPE_PERIODIC
try:
# We could do any of self.splitting_sequence(), self.canonical(), ...
# but this involves the least calculation and so is fastest.
self.splitting_sequences()
except flipper.AssumptionError:
return NT_TYPE_REDUCIBLE
return NT_TYPE_PSEUDO_ANOSOV
[docs] def is_abelian(self):
''' Return if this mapping class corresponds to an Abelian differential.
This is an Abelian differential (rather than a quadratic differential) if and
only if its stable lamination is orientable.
Assumes (and checks) that the mapping class is pseudo-Anosov.
This encoding must be a mapping class. '''
return self.splitting_sequence().lamination.is_orientable()
[docs] def is_primitive(self):
''' Return if this mapping class is primitive.
This encoding must be a mapping class.
Assumes (and checks) that this mapping class is pseudo-Anosov. '''
return self.splitting_sequences(take_roots=True).open_periodic.flip_length() == self.canonical().flip_length()
[docs] def is_conjugate_to(self, other):
''' Return if this mapping class is conjugate to other.
It would also be straightforward to check if self^i ~~ other^j
for some i, j.
Both encodings must be mapping classes.
Currently assumes that at least one mapping class is pseudo-Anosov. '''
assert isinstance(other, Encoding)
# Nielsen-Thurston type is a conjugacy invariant.
if self.nielsen_thurston_type() != other.nielsen_thurston_type():
return False
if self.nielsen_thurston_type() == NT_TYPE_PERIODIC:
if self.order() != other.order():
return False
# We could also use action on H_1(S) as a conjugacy invaraiant.
raise flipper.AssumptionError('Mapping class is periodic.')
elif self.nielsen_thurston_type() == NT_TYPE_REDUCIBLE:
# There's more to do here.
raise flipper.AssumptionError('Mapping class is reducible.')
else: # if self.nielsen_thurston_type() == NT_TYPE_PSEUDO_ANOSOV:
# Two pseudo-Anosov mapping classes are conjugate if and only if
# there canonical forms are cyclically conjugate via an isometry.
f = self.canonical()
g = other.canonical()
# We should start by quickly checking some invariants.
# For example they should have the same dilatation.
if self.dilatation() != other.dilatation():
return False
for i in range(len(f)):
# Conjugate around.
f_cycled = f[i:] * f[:i]
# g_cycled = g[i:] * g[:i] # Could cycle g instead.
for isom in f_cycled.source_triangulation.isometries_to(g.source_triangulation):
if isom.encode() * f_cycled == g * isom.encode():
return True
return False
[docs] def stratum(self):
''' Return a dictionary mapping each singularity to its stratum order.
This is the number of bipods incident to the vertex.
Assumes (and checks) that this mapping class is pseudo-Anosov.
This encoding must be a mapping class. '''
# This can fail with an flipper.AssumptionError.
return self.splitting_sequence().lamination.stratum()
[docs] def hitting_matrix(self):
''' Return the hitting matrix of the underlying train track. '''
# This can fail with an flipper.AssumptionError.
h = self.canonical()
M = flipper.kernel.id_matrix(h.zeta)
lamination = h.invariant_lamination()
# Lamination defines a train track with a bipod in each triangle. We
# follow the sequence of folds (and isometries) which this train track
# undergoes and track how the edges are mapped using M.
for item in reversed(h.sequence):
if isinstance(item, flipper.kernel.EdgeFlip):
triangulation = lamination.triangulation
a, b, c, d = triangulation.square_about_edge(item.edge_label)
# Work out which way the train track is pointing.
if lamination.is_bipod(triangulation.corner_of_edge(a.label)):
assert lamination.is_bipod(triangulation.corner_of_edge(c.label))
M = M.elementary(item.edge_index, b.index)
M = M.elementary(item.edge_index, d.index)
elif lamination.is_bipod(triangulation.corner_of_edge(b.label)):
assert lamination.is_bipod(triangulation.corner_of_edge(d.label))
M = M.elementary(item.edge_index, a.index)
M = M.elementary(item.edge_index, c.index)
else:
raise flipper.FatalError('Incompatible bipod.')
elif isinstance(item, flipper.kernel.Isometry):
M = flipper.kernel.Permutation([item.index_map[i] for i in range(h.zeta)]).matrix() * M
else:
raise flipper.FatalError('Unknown item in canonical sequence.')
# Move the lamination onto the next triangulation.
lamination = item(lamination)
return M
[docs] def bundle(self, veering=True, _safety=True):
''' Return the bundle associated to this mapping class.
This method can be run in two different modes:
If veering=True then the bundle returned is triangulated by a veering,
layered triangulation and has at most 6g+5n-6 additional loops drilled
from it, as described by Agol. These additional cusps are marked as
fake cusps and can be dealt with by filling along their fibre slope.
Assumes (and checks) that this mapping class is pseudo-Anosov.
If veering=False then the bundle returned is triangulated by a layered
triangulation obtained by stacking flat tetrahedra, one for each edge
flip in self.
Assumes (and checks) that the resulting triangulation is an ideal
triangulation of a manifold and that the fibre surface immerses into
the two skeleton. If _safety=True then this should always happen.
This encoding must be a mapping class. '''
assert self.is_mapping_class()
triangulation = self.source_triangulation
if veering:
# This can fail with an flipper.AssumptionError if self is not pseudo-Anosov.
return self.canonical().bundle(veering=False, _safety=False)
if _safety:
# We should add enough flips to ensure the triangulation is a manifold.
# Flipping and then unflipping every edge is certainly enough.
# However, we still have to be careful as there may be non-flippable edges.
# Start by adding a flip and unflip each flippable edge.
safe_encoding = self
for i in triangulation.flippable_edges():
extra = triangulation.encode_flip(i)
safe_encoding = extra.inverse() * extra * safe_encoding
# Then add a flip and unflip for each non-flippable edge.
for i in triangulation.indices:
if not triangulation.is_flippable(i):
# To do this we must first flip the boundary edge.
boundary_edge = triangulation.nonflippable_boundary(i)
# This edge is always flippable and, after flipping it, i is too.
extra = triangulation.encode([i, boundary_edge])
safe_encoding = extra.inverse() * extra * safe_encoding
return safe_encoding.bundle(veering=False, _safety=False)
id_perm3 = flipper.kernel.Permutation((0, 1, 2))
lower_triangulation, upper_triangulation = triangulation, triangulation
triangulation3 = flipper.kernel.Triangulation3(self.flip_length())
# These are maps taking triangles of lower (respectively upper) triangulation to either:
# - A pair (triangle, permutation) where triangle is in upper (resp. lower) triangulation, or
# - A pair (tetrahedron, permutation) of triangulation3.
# We start with no tetrahedra, so these maps are just the identity map between the two triangulations.
lower_map = dict((triangleA, (triangleB, id_perm3)) for triangleA, triangleB in zip(lower_triangulation, upper_triangulation))
upper_map = dict((triangleB, (triangleA, id_perm3)) for triangleA, triangleB in zip(lower_triangulation, upper_triangulation))
# We also use these two functions to quickly tell what a triangle maps to.
maps_to_triangle = lambda X: isinstance(X[0], flipper.kernel.Triangle)
maps_to_tetrahedron = lambda X: not maps_to_triangle(X)
tetra_count = 0
for item in reversed(self.sequence):
assert item.source_triangulation == upper_triangulation
try:
tetra_count, upper_triangulation, upper_map, lower_map = \
item.extend_bundle(triangulation3, tetra_count, upper_triangulation, lower_triangulation, upper_map, lower_map)
except AttributeError as err:
# We have no way to handle any other type that appears.
# Currently this means there was a LinearTransform and so this is not a mapping class.
raise flipper.FatalError('Unknown move %s encountered while building bundle.' % item) from err
# We're now back to the starting triangulation.
assert lower_triangulation == upper_triangulation
# This is a map which send each triangle of upper_triangulation via isometry to a pair:
# (triangle, permutation)
# where triangle in lower_triangulation and maps_to_tetrahedron(lower_map[triangle]).
full_forwards = dict()
for source_triangle in upper_triangulation:
target_triangle, perm = source_triangle, id_perm3
c = 0
while maps_to_triangle(lower_map[target_triangle]):
target_triangle, new_perm = lower_map[target_triangle]
perm = new_perm * perm
c += 1
assert c <= 3 * upper_triangulation.zeta
full_forwards[source_triangle] = (target_triangle, perm)
# Now close the bundle up.
for source_triangle in upper_triangulation:
if maps_to_tetrahedron(upper_map[source_triangle]):
A, perm_A = upper_map[source_triangle]
target_triangle, perm = full_forwards[source_triangle]
B, perm_B = lower_map[target_triangle]
A.glue(perm_A(3), B, perm_B * perm.embed(4) * perm_A.inverse())
# There are now no unglued faces.
assert triangulation3.is_closed()
# Install longitudes and meridians. This also calls Triangulation3.assign_cusp_indices().
triangulation3.install_peripheral_curves()
# Construct an immersion of the fibre surface into the closed bundle.
fibre_immersion = dict()
for source_triangle in lower_triangulation:
if maps_to_triangle(lower_map[source_triangle]):
upper_triangle, upper_perm = lower_map[source_triangle]
target_triangle, perm = full_forwards[upper_triangle]
B, perm_B = lower_map[target_triangle]
fibre_immersion[source_triangle] = (B, perm_B * (perm * upper_perm).embed(4))
else:
B, perm_B = lower_map[source_triangle]
fibre_immersion[source_triangle] = lower_map[source_triangle]
return flipper.kernel.Bundle(lower_triangulation, triangulation3, fibre_immersion)
def __snappy__(self):
return self.bundle(veering=False).snappy_string()
[docs] def flat_structure(self):
''' Return the flat structure associated to self.canonical().
This is based off of code supplied by Shannon Horrigan.
Assumes that this mapping class is pseudo-Anosov.
This encoding must be a mapping class. '''
assert self.is_mapping_class()
splitting_sequence = self.splitting_sequence()
periodic_triangulation = splitting_sequence.triangulation # This is the triangulation we will build the flat structure on.
stable_lamination = splitting_sequence.lamination # These give us x coordinates.
unstable_lamination = splitting_sequence.mapping_class.inverse().invariant_lamination() # These give us y coordinates.
edge_vectors = dict()
for triangle in periodic_triangulation:
# Find the sides with largest stable and unstable lengths.
index_s = max(range(3), key=lambda i: stable_lamination(triangle.edges[i])) # pylint: disable=cell-var-from-loop
index_u = max(range(3), key=lambda i: unstable_lamination(triangle.edges[i])) # pylint: disable=cell-var-from-loop
# Get the edges of triangle relative to the index_s.
edges = [triangle[(index_s + i) % 3] for i in range(3)]
if (index_s + 1) % 3 == index_u: # If the longest stable side is followed by the longest unstable side.
edge_vectors[edges[0]] = flipper.kernel.Vector2(+stable_lamination(edges[0]), -unstable_lamination(edges[0]))
edge_vectors[edges[1]] = flipper.kernel.Vector2(-stable_lamination(edges[1]), +unstable_lamination(edges[1]))
edge_vectors[edges[2]] = flipper.kernel.Vector2(-stable_lamination(edges[2]), -unstable_lamination(edges[2]))
elif (index_s - 1) % 3 == index_u: # If the longest unstable side is followed by the longest stable side.
edge_vectors[edges[0]] = flipper.kernel.Vector2(+stable_lamination(edges[0]), +unstable_lamination(edges[0]))
edge_vectors[edges[1]] = flipper.kernel.Vector2(-stable_lamination(edges[1]), +unstable_lamination(edges[1]))
edge_vectors[edges[2]] = flipper.kernel.Vector2(-stable_lamination(edges[2]), -unstable_lamination(edges[2]))
else: # Longest stable and unstable sides are the same.
raise flipper.FatalError('Longest stable and unstable edges are the same.')
assert sum([edge_vectors[edge] for edge in triangle], flipper.kernel.Vector2(0, 0)) == flipper.kernel.Vector2(0, 0)
return flipper.kernel.FlatStructure(periodic_triangulation, edge_vectors)
def create_encoding(source_triangulation, sequence, _cache=None):
''' Return the encoding defined by sequence starting at source_triangulation.
This is only really here to help with pickling. Users should use
source_triangulation.encode(sequence) directly. '''
assert isinstance(source_triangulation, flipper.kernel.Triangulation)
return source_triangulation.encode(sequence, _cache=_cache)
