def exotic_angles(self):
"""Given a transverse taut veering triangulation, find the exotic taut structures (as defined in Futer-Gueritaud)"""
### From Futer-Gueritaud's leading-trailing deformations, the exotic angles should be derivable even without
### a transverse-taut structure, but it seems easier with.
exotic_upper = []
exotic_lower = []
for i, tet in enumerate(self.tri.tetrahedra()):
top_edge, bottom_edge = get_tet_top_and_bottom_edges(
self.coorientations, tet)
top_colour, bottom_colour = self.veering_colours[
top_edge.index()], self.veering_colours[bottom_edge.index()]
tet_angle = self.angle[i]
equator = (tet_angle + 1) % 3
equator_colour = self.veering_colours[tet.edge(equator).index()]
assert self.veering_colours[tet.edge(
5 - equator).index()] == equator_colour
if equator_colour == top_colour:
exotic_upper.append(equator)
else:
exotic_upper.append((equator + 1) % 3)
if equator_colour == bottom_colour:
exotic_lower.append(equator)
else:
exotic_lower.append((equator + 1) % 3)
assert is_taut(self.tri, exotic_upper) and is_taut(
self.tri, exotic_lower)
return [exotic_upper, exotic_lower]
def num_veering_structs(M, angles=None, use_flipper=True):
"""
Tries to count them (in a very naive way).
Tries to eliminate overcounting due to symmetries (in reduced_angles)
but will fail if one of the symmetries is hidden by retriangulation.
If use_flipper = False then we get a (true) lower bound, since then it only
counts veering structures on the given triangulation.
"""
if angles == None:
angles = reduced_angles(M)
tri = regina.Triangulation3(M)
for angle in angles:
if not is_taut(tri, angle):
return None
for angle in angles:
if not (is_veering(tri, angle) or is_layered(tri, angle)):
return None
total = 0
for angle in angles:
if is_veering(tri, angle):
total = total + 1
elif use_flipper:
assert is_layered(tri, angle)
print(M.name(), angle, "needs flipper")
try:
if not has_internal_singularities(tri, angle):
total = total + 1
except:
print("flipper failed")
return total
def mutate(tri,
angle,
weights,
isom,
tet_vert_coorientations=None,
quiet=False):
if tet_vert_coorientations == None:
tet_vert_coorientations = is_transverse_taut(
tri, angle, return_type="tet_vert_coorientations")
regluing = surface_isom_to_regluing_pattern(tri, angle, weights, isom,
tet_vert_coorientations)
#print('regluing data:', regluing)
# first unglue all faces that need to be unglued, then reglue them via isom
# can't do both at the same step, because regina goes crazy when you try to glue a face to a face which is already glued somewhere else
for x in regluing:
tet_below_top_triangle = tri.tetrahedron(x[0])
which_face = x[1]
tet_below_top_triangle.unjoin(which_face)
#print('unglued face', which_face, 'of tet', tet_below_top_triangle.index())
for x in regluing:
tet_below_top_triangle = tri.tetrahedron(x[0])
which_face = x[1]
tet_above_image_triangle = tri.tetrahedron(x[2])
perm = x[3]
tet_below_top_triangle.join(which_face, tet_above_image_triangle, perm)
#print('glued face', which_face, 'of tet', tet_below_top_triangle.index(), 'to tet', tet_above_image_triangle.index(), 'via', perm)
assert tri.isValid()
assert tri.countBoundaryFacets() == 0
if quiet == False:
print('mutant isosig:', tri.isoSig())
print('taut:', is_taut(tri, angle))
if is_taut(tri, angle):
print('transverse taut:', is_transverse_taut(tri, angle))
print('veering:', is_veering(tri, angle))
else:
edge_num = tri.countEdges()
print('Got', edge_num, 'edges out of', tri.countTetrahedra())
totals = is_taut(tri, angle, return_totals=True)
print('Angles:', totals)
def __init__(self, tri, angle, tet_shapes=None):
self.tri = tri
self.angle = angle
assert is_taut(tri, angle)
self.coorientations = is_transverse_taut(
tri, angle, return_type="tet_vert_coorientations")
assert self.coorientations != False
self.veering_colours = is_veering(tri,
angle,
return_type="veering_colours")
assert self.veering_colours != False
self.tet_types = is_veering(tri, angle, return_type="tet_types")
self.tet_shapes = tet_shapes
def can_deal_with_reduced_angle(tri, angle):
"""
Returns True or False, as our techniques can recognise the given
reduced angle structure.
"""
if not is_taut(tri, angle):
return False
if is_veering(tri, angle):
return True
if is_layered(tri,
angle): # has_internal_singularities is not needed here.
return True
return False
def is_transverse_taut(triangulation,
taut_angle_struct,
return_type="boolean"):
assert is_taut(triangulation, taut_angle_struct)
assert triangulation.isOriented()
coorientations = []
for i in range(triangulation.countTetrahedra()):
coorientations.append([0, 0, 0, 0]) # +1 is out of tet, -1 is into tet
# first decide coorientation for tet 0
edgepair = taut_angle_struct[0]
if not add_coors_with_check(triangulation, coorientations, 0, edgepair,
+1): # choose arbitrary direction for tet 0
return False # tet 0 could have self gluings incompatible with the transverse taut structure, so we could fail here
explored_tetrahedra = [0]
frontier_tet_faces = [(0, 0), (0, 1), (0, 2), (0, 3)]
while len(frontier_tet_faces) > 0:
my_tet_num, my_face_num = frontier_tet_faces.pop()
my_tet = triangulation.tetrahedron(my_tet_num)
neighbour_tet = my_tet.adjacentSimplex(my_face_num)
neighbour_face_num = my_tet.adjacentGluing(my_face_num)[my_face_num]
if neighbour_tet.index() in explored_tetrahedra:
frontier_tet_faces.remove(
(neighbour_tet.index(), neighbour_face_num))
else:
newcoor = -coorientations[my_tet_num][
my_face_num] # opposite of the adjacent face's coorientation
edgepair = taut_angle_struct[neighbour_tet.index()]
# now calculate the direction that the flow points through neighbour_tet
if neighbour_face_num in [
vertexSplit[edgepair][i] for i in range(2)
]:
direction = newcoor # agrees with newcoor
else:
direction = -newcoor
if not add_coors_with_check(
triangulation, coorientations, neighbour_tet.index(),
taut_angle_struct[neighbour_tet.index()], direction):
return False
frontier_tet_faces.extend([(neighbour_tet.index(), i)
for i in range(4)
if i != neighbour_face_num])
explored_tetrahedra.append(neighbour_tet.index())
if return_type == "tet_vert_coorientations":
return coorientations
elif return_type == "face_coorientations":
return convert_tetrahedron_coorientations_to_faces(
triangulation, coorientations)
else:
return True
# sig = 'eLMkbcddddedde_2100'
# sig = 'eLMkbcdddhxqlm_1200'
# sig = 'eLAkaccddjsnak_2001'
# sig = 'eLAkbbcdddhwqj_2102'
# sig = 'fLLQcbcdeeelonlel_02211' ## should have three different midsurface bdy components, so four total after drilling
# sig = 'fLLQcbddeeehrcdui_12000' ## five boundary components
# sig = 'fLAMcbbcdeedhwqqs_21020'
# sig = 'qvvLPQwAPLQkfhgffijlkmnopoppoaaaaaaaaaaaaadddd_1020212200111100'
# t = excise_fans(sig)
# t.save('excise_fans_' + sig + '.rga')
# t0, _ = isosig_to_tri_angle('cPcbbbdxm_10')
# t1, _ = isosig_to_tri_angle('cPcbbbiht_12')
# print t.isIsomorphicTo(t0) != None or t.isIsomorphicTo(t1) != None
# print fan_stacks(sig)
import veering
import transverse_taut
import taut
from file_io import parse_data_file
census = parse_data_file('Data/veering_census.txt')
for sig in census[:200]:
tri, angle = excise_fans(sig)
print((sig, tri.countTetrahedra(), angle, taut.is_taut(tri, angle)))
assert veering.is_veering(tri, angle)
assert transverse_taut.is_transverse_taut(tri, angle)
# print sig, fan_stacks(sig)
def drill(tri,
loop,
angle=None,
branch=None,
sig=None): # sig just for diagnostics
"""
Returns the new cusp formed by drilling
"""
if angle != None:
face_coorientations = is_transverse_taut(
tri, angle, return_type="face_coorientations")
assert face_coorientations != False
tet_vert_coorientations = is_transverse_taut(
tri, angle, return_type="tet_vert_coorientations")
assert tet_vert_coorientations != False
original_tri = regina.Triangulation3(tri)
original_countTetrahedra = tri.countTetrahedra()
original_countBoundaryComponents = tri.countBoundaryComponents()
### add new tetrahedra
new_0_tets = []
new_1_tets = [] ## both relative to regina's two embeddings for the face
for i in range(len(loop)):
new_0_tets.append(tri.newTetrahedron())
new_1_tets.append(tri.newTetrahedron())
### we will glue tetrahedra together with a numbering that is convenient but
### unfortunately not oriented. We will orient later.
# 1 pivot
# /|\
# / | \
# / ,3. \
# /,' `.\
# 0---------2 leading
# trailing
for i in range(len(loop)):
new_0_tets[i].join(1, new_1_tets[i], regina.Perm4(0, 1, 2, 3))
### now glue along the loop path, need to worry about regina's embeddings for neighbouring triangles
loop_face_tets0 = []
loop_face_vertices0 = []
loop_face_tets1 = []
loop_face_vertices1 = [
] ### need to store these because they cannot be recomputed once we start ungluing faces
for i in range(len(loop)):
# print('collect info: i', i)
face_data = loop[i]
face_index = face_data[0]
vert_nums = face_data[1]
face = tri.triangles()[face_index]
face_embed0 = face.embedding(0)
face_tet = face_embed0.simplex()
face_vertices = face_embed0.vertices()
face_opposite_vert = face_vertices[3]
face_other_non_edge_vert = face_vertices[
vert_nums[0]] ## opposite trailing vertex
face_data_next = loop[(i + 1) % len(loop)]
face_index_next = face_data_next[0]
vert_nums_next = face_data_next[1]
face_next = tri.triangles()[face_index_next]
face_next_embed0 = face_next.embedding(0)
face_next_tet = face_next_embed0.simplex()
face_next_vertices = face_next_embed0.vertices()
face_next_opposite_vert = face_next_vertices[3]
face_next_other_non_edge_vert = face_next_vertices[
vert_nums_next[2]] ## opposite leading vertex
### things to store for later
loop_face_tets0.append(face_tet) # store for later
loop_face_vertices0.append(face_vertices) #store for later
face_embed1 = face.embedding(1)
loop_face_tets1.append(face_embed1.simplex())
loop_face_vertices1.append(face_embed1.vertices())
edge = face.edge(vert_nums[0]) ## opposite trailing vertex
# print('edge index', edge.index())
assert edge == face_next.edge(
vert_nums_next[2]) ## opposite leading vertex
edgemapping = face.faceMapping(1, vert_nums[0])
next_edgemapping = face_next.faceMapping(1, vert_nums_next[2])
face_gluing_regina_numbering = next_edgemapping * (
edgemapping.inverse()
) ### maps vertices 0,1,2 on face to corresponding vertices on face_next
assert face_gluing_regina_numbering[3] == 3
face_gluing_regina_numbering = regina.Perm3(
face_gluing_regina_numbering[0], face_gluing_regina_numbering[1],
face_gluing_regina_numbering[2])
assert face_gluing_regina_numbering[vert_nums[0]] == vert_nums_next[2]
face_gluing = vert_nums_next.inverse(
) * face_gluing_regina_numbering * vert_nums
assert face_gluing[0] == 2
signs = []
edge_embeddings = edge.embeddings()
for embed in edge_embeddings:
if embed.simplex() == face_tet:
if set([embed.vertices()[2],
embed.vertices()[3]]) == set(
[face_opposite_vert, face_other_non_edge_vert]):
# print('embed data face_tet', embed.simplex().index(), embed.vertices())
# print('embed edge vert nums', embed.vertices()[0], embed.vertices()[1])
signs.append(face_opposite_vert == embed.vertices()[2])
if embed.simplex() == face_next_tet:
if set([embed.vertices()[2],
embed.vertices()[3]]) == set([
face_next_opposite_vert,
face_next_other_non_edge_vert
]):
# print('embed data face_next_tet', embed.simplex().index(), embed.vertices())
# print('embed edge vert nums', embed.vertices()[0], embed.vertices()[1])
signs.append(
face_next_opposite_vert == embed.vertices()[2])
# print('signs', signs)
assert len(signs) == 2
if signs[0] == signs[
1]: ### coorientations are same around the edge (not a transverse taut coorientation!)
new_0_tets[i].join(
0, new_1_tets[(i + 1) % len(loop)],
regina.Perm4(2, face_gluing[1], face_gluing[2], 3))
new_1_tets[i].join(
0, new_0_tets[(i + 1) % len(loop)],
regina.Perm4(2, face_gluing[1], face_gluing[2], 3))
else:
new_0_tets[i].join(
0, new_0_tets[(i + 1) % len(loop)],
regina.Perm4(2, face_gluing[1], face_gluing[2], 3))
new_1_tets[i].join(
0, new_1_tets[(i + 1) % len(loop)],
regina.Perm4(2, face_gluing[1], face_gluing[2], 3))
### now unglue tri along the loop and glue in the new tetrahedra
for i in range(len(loop)):
# print('modify triangulation: i', i)
vert_nums = loop[i][1]
face_tet0 = loop_face_tets0[i]
face_vertices0 = loop_face_vertices0[i]
face_tet1 = loop_face_tets1[i]
face_vertices1 = loop_face_vertices1[i]
face_opposite_vert0 = face_vertices0[3]
face_tet0.unjoin(face_opposite_vert0)
### glue torus shell to the old tetrahedra
vert_nums_Perm4 = regina.Perm4(vert_nums[0], vert_nums[1],
vert_nums[2], 3)
new_0_tets[i].join(3, face_tet0, face_vertices0 * vert_nums_Perm4)
new_1_tets[i].join(3, face_tet1, face_vertices1 * vert_nums_Perm4)
assert tri.isValid()
assert tri.countBoundaryComponents(
) == original_countBoundaryComponents + 1
if angle != None:
for i in range(len(loop)):
face_data = loop[i]
face_index = face_data[0]
face = original_tri.triangles()[face_index]
vert_nums = face_data[1]
face_embed0 = face.embedding(0)
face_tet = face_embed0.simplex()
face_vertices = face_embed0.vertices()
face_opposite_vert = face_vertices[3]
coor_points_out_of_tet0 = (tet_vert_coorientations[
face_tet.index()][face_opposite_vert] == +1)
# if (flow_agrees_with_regina_numbers != face_cor_agrees_with_regina_numbers) != coor_points_out_of_tet0:
if coor_points_out_of_tet0:
angle.extend([0, 2])
else:
angle.extend([2, 0])
# print(sig, loop, angle, is_taut(tri, angle))
assert is_taut(tri, angle)
if branch != None:
for i in range(len(loop)):
branch.extend([1, 1])
assert is_branched(tri, branch)
M = snappy.Manifold(tri)
if M.volume() < 1.0:
print('not hyperbolic', sig, loop, angle, M.volume())
assert False
# print(M.verify_hyperbolicity()) ### very slow
# print(M.volume())
# assert M.volume() > 1.0 ###
### now orient
swaps = [regina.Perm4()
] * original_countTetrahedra ### identity permutations
for i in range(len(loop)):
if new_0_tets[i].adjacentGluing(3).sign() == 1:
if angle != None:
this_tet_angle = angle[original_countTetrahedra + 2 * i]
else:
this_tet_angle = 0 ### pi_location = 0 is an arbitrary choice
swaps.append(
reverse_tet_orientation(tri, new_0_tets[i], this_tet_angle))
else:
swaps.append(regina.Perm4())
if new_1_tets[i].adjacentGluing(3).sign() == 1:
if angle != None:
this_tet_angle = angle[original_countTetrahedra + 2 * i + 1]
else:
this_tet_angle = 0 ### pi_location = 0 is an arbitrary choice
swaps.append(
reverse_tet_orientation(tri, new_1_tets[i], this_tet_angle))
else:
swaps.append(regina.Perm4())
assert tri.isOriented()
if angle != None:
assert is_taut(tri, angle)
if branch != None:
# print('loop, branch, swaps', loop, branch, swaps)
apply_swaps_to_branched_surface(branch, swaps)
# print('loop, branch, swaps', loop, branch, swaps)
assert is_branched(tri, branch)
assert has_non_sing_semiflow(tri, branch)
### return the vertex of the triangulation corresponding to the drilled cusp
drilled_cusp = new_0_tets[0].vertex(swaps[new_0_tets[0].index()][3])
assert drilled_cusp.degree() == len(new_0_tets) + len(new_1_tets)
return drilled_cusp.index()
def veering_mobius_dehn_surgery(triangulation, angle_struct, face_num):
tri = regina.Triangulation3(triangulation) # make a copy
angle = list(angle_struct) # make a copy
face = tri.triangle(face_num)
assert face.isMobiusBand()
# Note that dunce caps cannot appear in a veering triangulation
# Find which vertex is on both copies of the identified edge of the face
edges = [face.edge(i)
for i in range(3)] # edge i is opposite vertex i, i in [0, 1, 2]
for j in range(3):
if edges[j] == edges[(j + 1) % 3]:
B = (j + 2) % 3
break
embed0 = face.embedding(0)
embed1 = face.embedding(1)
tet0 = embed0.tetrahedron()
tet1 = embed1.tetrahedron()
embed0_verts = embed0.vertices()
embed1_verts = embed1.vertices()
# In tet0: B gives "b". Let "c" be the edge sharing a
# pi with "b". Let "d" be the vertex not meeting the given face.
# Let "a" be the remaining vertex.
b = embed0.vertices()[B]
c = shares_pi_with(angle[tet0.index()], b)
d = embed0.vertices()[3] # ... use the face index
a = [i for i in [0, 1, 2, 3]
if i not in [b, c, d]].pop() # ... whatever is left
# similarly in tet1 - B gives "q". Let "r" be the edge sharing a
# pi with "q". Let "s" be the vertex not meeting the given face.
# Let "p" be the remaining vertex.
q = embed1.vertices()[B]
r = shares_pi_with(angle[tet1.index()], q)
s = embed1.vertices()[3] # ... use the face index
p = [i for i in [0, 1, 2, 3]
if i not in [q, r, s]].pop() # ... whatever is left
# get colour of mobius strip
pair_a = [b, c]
pair_a.sort()
mob_edge_a = tet0.edge(vert_pair_to_edge_num[tuple(pair_a)])
pair_c = [a, b]
pair_c.sort()
mob_edge_c = tet0.edge(vert_pair_to_edge_num[tuple(pair_c)])
assert mob_edge_a == mob_edge_c
veering_colours = is_veering(tri, angle, return_type="veering_colours")
assert veering_colours != False # otherwise the triangulation is not veering
mob_colour = veering_colours[mob_edge_a.index()]
# Now actually do the surgery
tet0.unjoin(d) # same as tet1.unjoin(s)
tet_new = tri.newTetrahedron()
if mob_colour == "red":
tet_new.join(0, tet_new, regina.Perm4(3, 0, 1, 2))
tet_new.join(1, tet0, regina.Perm4(c, d, a, b))
tet_new.join(2, tet1, regina.Perm4(q, p, s, r))
else:
tet_new.join(1, tet_new, regina.Perm4(1, 3, 0, 2))
tet_new.join(2, tet0, regina.Perm4(a, b, d, c))
tet_new.join(0, tet1, regina.Perm4(s, r, p, q))
angle.append(0) # this is the correct taut angle for our new tetrahedron
assert is_taut(tri, angle)
assert is_veering(tri, angle)
return tri, angle, tet_new.triangle(3).index()
def twoThreeMove(tri, angle, face_num, perform = True, return_edge = False):
"""Apply a 2-3 move to a taut triangulation, if possible.
If perform = False, returns if the move is possible.
If perform = True, modifies tri, returns (tri, angle) for the performed move"""
face = tri.triangle(face_num)
embed0 = face.embedding(0)
tet0 = embed0.simplex()
tet_num0 = tet0.index()
tet_0_face_num = embed0.face()
vertices0 = embed0.vertices() # Maps vertices (0,1,2) of face to the corresponding vertex numbers of tet0
embed1 = face.embedding(1)
tet1 = embed1.simplex()
tet_num1 = tet1.index()
tet_1_face_num = embed1.face()
vertices1 = embed1.vertices() # Maps vertices (0,1,2) of face to the corresponding vertex numbers of tet1
if tet0 == tet1: ### Cannot perform a 2-3 move across a self-gluing
return False
### taut 2-3 move is valid if the pis are on different edges of face
### this never happens if we start with a veering triangulation.
### for veering, the two-tetrahedron ball is always a continent.
for i in range(3):
j = (i+1) % 3
k = (i+2) % 3
if angle[tet_num0] == unsorted_vert_pair_to_edge_pair[(vertices0[j], vertices0[k])]:
pi_num_0 = i
if angle[tet_num1] == unsorted_vert_pair_to_edge_pair[(vertices1[j], vertices1[k])]:
pi_num_1 = i
if pi_num_0 == pi_num_1:
return False
if perform == False:
return True
### check we do the same as regina...
tri2 = regina.Triangulation3(tri) ## make a copy
tri2.pachner(tri2.triangle(face_num))
### We have to implement twoThreeMove ourselves. e.g. we do a 2-3 move to canonical fig 8 knot complement triangulation.
### All of the original tetrahedra are removed. I don't see any way to carry the angle structure through without knowing
### exactly how Ben's implementation works.
## record the tetrahedra and gluings adjacent to tet0 and tet1
tets = [tet0, tet1]
vertices = [vertices0, vertices1]
# print('2-3 vertices signs')
# print([v.sign() for v in vertices])
gluings = []
for i in range(2):
tet_gluings = []
for j in range(3):
tet_gluings.append( [ tets[i].adjacentTetrahedron(vertices[i][j]), tets[i].adjacentGluing(vertices[i][j])] )
# if tets[i].adjacentTetrahedron(vertices[i][j]) in tets:
# print('self gluing')
gluings.append(tet_gluings)
### add new tetrahedra
new_tets = []
for i in range(3):
new_tets.append(tri.newTetrahedron())
### glue around degree 3 edge
for i in range(3):
new_tets[i].join(2, new_tets[(i+1)%3], regina.Perm4(0,1,3,2))
### replace mapping info with corresponding info for the 3 tet. Self gluings will be annoying...
### write verticesi[j] as vij
### tet0 new_tet0
### _________ _________
### ,'\ /`. ,'\`. ,'/`.
### ,' \ v03 / `. ,' \ `0' / `.
### ,' \ / `. ,' \ | / `.
### / \ \ / / \ /|\ \|/ /|\
### /v02\ * /v01\ / | \ * / | \
### / _\..... | ...../_ \ / | 3\..... | ...../2 | \
### /_--"" / * \ ""--_\ /2 ,' / * \ `. 3\
### \`.v12/ / \ \v11,'/ `. \`.| / /|\ \ |,'/
### \ `./ / \ \,' / ----} \ `./ / | \ \,' /
### \ /`. / v00 \ ,'\ / ,' \|/`. / | \ ,'\|/
### \ `. / \ ,' / \ `. / | \ ,' /
### \ `---------' / \ * 3 | 2 * /
### \ \ / / \ \ | / /
### \ \ v10 / / new_tet1 \ \ | / / new_tet2
### \ \ / / \ \ | / /
### \ \ / / \ \|/ /
### \ * / \ * /
### tet1 \...|.../ \...|.../
### \ | / \`.|.'/
### \v13/ \ 1 /
### \|/ \|/
### * *
# permutations taking the vertices for a face of the 3-tet ball to the
# vertices of the same face for the 2-tet ball
# these should be even in order to preserve orientability.
# exactly one of vertices[0] and vertices[1] is even, but it seems to depend on the face.
# perms = [[regina.Perm4( vertices[0][3], vertices[0][0], vertices[0][1], vertices[0][2] ), ### opposite v00
# regina.Perm4( vertices[0][3], vertices[0][1], vertices[0][2], vertices[0][0] ), ### opposite v01
# regina.Perm4( vertices[0][3], vertices[0][2], vertices[0][0], vertices[0][1] ) ### opposite v02
# ],
# [regina.Perm4( vertices[1][0], vertices[1][3], vertices[1][1], vertices[1][2] ), ### opposite v10
# regina.Perm4( vertices[1][1], vertices[1][3], vertices[1][2], vertices[1][0] ), ### opposite v11
# regina.Perm4( vertices[1][2], vertices[1][3], vertices[1][0], vertices[1][1] ) ### opposite v12
# ]
# ]
perms = [[vertices[0] * regina.Perm4( 3,0,1,2 ), ### opposite v00
vertices[0] * regina.Perm4( 3,1,2,0 ), ### opposite v01
vertices[0] * regina.Perm4( 3,2,0,1 ) ### opposite v02
],
[vertices[1] * regina.Perm4( 0,3,1,2 ), ### opposite v10
vertices[1] * regina.Perm4( 1,3,2,0 ), ### opposite v11
vertices[1] * regina.Perm4( 2,3,0,1 ) ### opposite v12
]
]
flip = perms[0][0].sign() == -1
if flip: #then all of the signs are wrong, switch 0 and 1 on input
perms = [[p * regina.Perm4( 1,0,2,3 ) for p in a] for a in perms]
# print('2-3 perms signs')
# print([[p.sign() for p in a] for a in perms])
for i in range(2):
for j in range(3):
gluing = gluings[i][j]
if gluing != None:
if gluing[0] not in tets: ### not a self gluing
gluing[1] = gluing[1] * perms[i][j]
else:
i_other = tets.index( gluing[0] )
otherfacenum = gluing[1][vertices[i][j]]
j_other = [vertices[i_other][k] for k in range(4)].index(otherfacenum)
assert gluings[i_other][j_other][0] == tets[i]
assert gluings[i_other][j_other][1].inverse() == gluings[i][j][1]
gluings[i_other][j_other] = None ### only do a self gluing from one side
gluing[0] = new_tets[j_other]
gluing[1] = perms[i_other][j_other].inverse() * gluing[1] * perms[i][j]
### unglue two tetrahedra
tet0.isolate()
tet1.isolate()
### remove the tetrahedra
tri.removeSimplex(tet0)
tri.removeSimplex(tet1)
### make the gluings on the boundary of the new ball
for i in range(2):
for j in range(3):
if gluings[i][j] != None:
if flip:
new_tets[j].join(i, gluings[i][j][0], gluings[i][j][1])
else:
new_tets[j].join(1 - i, gluings[i][j][0], gluings[i][j][1])
assert tri.isIsomorphicTo(tri2)
assert tri.isOriented()
### update the angle structure
tet_indices = [tet_num0, tet_num1]
tet_indices.sort()
angle.pop(tet_indices[1])
angle.pop(tet_indices[0]) ## remove from the list in the correct order!
new_angle = [None, None, None]
new_angle[pi_num_0] = 0
new_angle[pi_num_1] = 0 ### these two tetrahedra have their pi's on the new degree three edge
third_index = 3 - (pi_num_0 + pi_num_1)
if (pi_num_0 - third_index) % 3 == 1:
new_angle[third_index] = 1
else:
assert (pi_num_0 - third_index) % 3 == 2
new_angle[third_index] = 2
if flip:
new_angle[third_index] = 3 - new_angle[third_index]
angle.extend(new_angle)
assert is_taut(tri, angle)
if not return_edge:
return [ tri, angle ]
else:
return [ tri, angle, new_tets[0].edge(0).index() ]
def threeTwoMove(tri, angle, edge_num, perform = True, return_triangle = False):
"""Apply a 3-2 move to a taut triangulation, if possible.
If perform = False, returns if the move is possible.
If perform = True, modifies tri, returns (tri, angle) for the performed move"""
edge = tri.edge(edge_num)
if edge.degree() != 3:
return False
tets = []
tet_nums = []
vertices = []
non_pi_tet_num = None
for i in range(3):
embed = edge.embedding(i)
tets.append(embed.simplex())
tet_nums.append(tets[i].index())
vertices.append(embed.vertices())
if not there_is_a_pi_here(angle, embed):
assert non_pi_tet_num == None
non_pi_tet_num = embed.simplex().index()
local_non_pi_tet_num = i
if len(set([tet.index() for tet in tets])) != 3:
return False ### tetrahedra must be distinct
if not perform:
return True ### taut 3-2 move is always possible if the 3-2 move is.
### record the "slope" of the pis on the non_pi_tet. This is a boolean
non_pi_tet_positive = unsorted_vert_pair_to_edge_pair[ ( vertices[local_non_pi_tet_num][0], vertices[local_non_pi_tet_num][2] ) ]
is_positive_slope = (angle[non_pi_tet_num] == non_pi_tet_positive)
### check we do the same as regina...
tri2 = regina.Triangulation3(tri) ## make a copy
tri2.pachner(tri2.edge(edge_num))
## record the tetrahedra and gluings adjacent to the tets
gluings = []
for i in range(3):
tet_gluings = []
for j in range(2):
tet_gluings.append( [ tets[i].adjacentTetrahedron(vertices[i][j]), tets[i].adjacentGluing(vertices[i][j])] )
gluings.append(tet_gluings)
for i in range(3):
assert tets[i].adjacentTetrahedron(vertices[i][2]) == tets[(i+1)%3] ### The edge embeddings should be ordered this way...
### add new tetrahedra
new_tets = []
for i in range(2):
new_tets.append(tri.newTetrahedron())
### glue across face
new_tets[0].join(3, new_tets[1], regina.Perm4(0,2,1,3))
### replace mapping info with corresponding info for the 2 tet. Self gluings will be annoying...
### write vertices[i][j] as vij
### tets[0] new_tet1
### _________ _________
### ,'\`.v00,'/`. ,'\ /`.
### ,' \ `.' / `. ,' \ 3 / `.
### ,' v10\ | /v20 `. ,' \ / `.
### /|\ \|/ /|\ / \ \ / / \
### / | \ * / | \ / \ * / \
### v12/ | \..... | ...../ | \v23 / 1 _\..... | ...../_ 2 \
### / ,' / * \ `. \ /_--"" / * \ ""--_\
### \`.| /v03 /|\ v02\ |,'/ `. \`. 2 / / \ \ 1 ,'/
### \ `./ / | \ \,' / ----} \ `./ / \ \,' /
### \|/`. / | \ ,'\|/ ,' \ /`. / 0 \ ,'\ /
### \ `. / | \ ,' / \ `. / \ ,' /
### \ * v13|v22 * / \ `---------' /
### \ \ | / / \ \ / /
### \ \ | / / \ \ 0 / /
### \ \ | / / \ \ / /
### tets[1] \ \|/ / tets[2] \ \ / /
### \ * / \ * / new_tet0
### \..v01../ \...|.../
### \`.|.'/ \ | /
### v11\ | /v21 \ 3 /
### \|/ \|/
### * *
# permutations taking the vertices for a face of the 2-tet ball to the
# vertices of the same face for the 3-tet ball
# these should be even in order to preserve orientability.
# perms = [[regina.Perm4( vertices[0][0], vertices[0][2], vertices[0][3], vertices[0][1] ), ### opposite v00
# regina.Perm4( vertices[0][1], vertices[0][3], vertices[0][2], vertices[0][0] ) ### opposite v01
# ],
# [regina.Perm4( vertices[1][3], vertices[1][0], vertices[1][2], vertices[1][1] ), ### opposite v10
# regina.Perm4( vertices[1][3], vertices[1][2], vertices[1][1], vertices[1][0] ) ### opposite v11
# ],
# [regina.Perm4( vertices[2][2], vertices[2][3], vertices[2][0], vertices[2][1] ), ### opposite v20
# regina.Perm4( vertices[2][2], vertices[2][1], vertices[2][3], vertices[2][0] ) ### opposite v21
# ]
# ]
perms = [[vertices[0] * regina.Perm4( 0, 2, 3, 1 ), ### opposite v00
vertices[0] * regina.Perm4( 1, 3, 2, 0 ) ### opposite v01
],
[vertices[1] * regina.Perm4( 3, 0, 2, 1 ), ### opposite v10
vertices[1] * regina.Perm4( 3, 2, 1, 0 ) ### opposite v11
],
[vertices[2] * regina.Perm4( 2, 3, 0, 1 ), ### opposite v20
vertices[2] * regina.Perm4( 2, 1, 3, 0 ) ### opposite v21
]
]
for i in range(3):
for j in range(2):
gluing = gluings[i][j]
if gluing != None:
if gluing[0] not in tets: ### not a self gluing
gluing[1] = gluing[1] * perms[i][j]
else:
i_other = tets.index( gluing[0] )
otherfacenum = gluing[1][vertices[i][j]]
j_other = [vertices[i_other][k] for k in range(4)].index(otherfacenum)
assert gluings[i_other][j_other][0] == tets[i]
assert gluings[i_other][j_other][1].inverse() == gluings[i][j][1]
gluings[i_other][j_other] = None ### only do a self gluing from one side
gluing[0] = new_tets[j_other] ### j refers to the vertex on the same 3 side
gluing[1] = perms[i_other][j_other].inverse() * gluing[1] * perms[i][j]
### unglue three tetrahedra
for tet in tets:
tet.isolate()
### remove the tetrahedra
for tet in tets:
tri.removeSimplex(tet)
### make the gluings on the boundary of the new ball
for i in range(3):
for j in range(2):
if gluings[i][j] != None:
if j == 0 or i == 0:
assert new_tets[j].adjacentTetrahedron(i) == None ## not glued
assert gluings[i][j][0].adjacentTetrahedron(gluings[i][j][1][i]) == None
new_tets[j].join(i, gluings[i][j][0], gluings[i][j][1])
else:
assert new_tets[j].adjacentTetrahedron(3 - i) == None ## not glued
assert gluings[i][j][0].adjacentTetrahedron(gluings[i][j][1][3 - i]) == None
new_tets[j].join(3 - i, gluings[i][j][0], gluings[i][j][1]) ## swap 1 and 2
assert tri.isIsomorphicTo(tri2)
assert tri.isOriented()
# ### update the angle structure
tet_nums.sort()
angle.pop(tet_nums[2])
angle.pop(tet_nums[1])
angle.pop(tet_nums[0]) ## remove from the list in the correct order!
if local_non_pi_tet_num == 0:
if is_positive_slope:
new_angle = [0, 0]
else:
new_angle = [1, 1]
elif local_non_pi_tet_num == 1:
if is_positive_slope:
new_angle = [2, 1]
else:
new_angle = [0, 2]
else:
assert local_non_pi_tet_num == 2
if is_positive_slope:
new_angle = [1, 2]
else:
new_angle = [2, 0]
angle.extend(new_angle)
assert is_taut(tri, angle)
if not return_triangle:
return (tri, angle)
else:
return (tri, angle, new_tets[0].triangle(3).index())
def threeTwoMove(tri, edge_num, angle = None, branch = None, perform = True, return_triangle = False, return_vertex_perm = False):
"""Apply a 3-2 move to a triangulation with a taut structure and/or branched surface, if possible.
If perform = False, returns if the move is possible.
modifies tri, returns (tri, angle, branch) for the performed move.
If return_vertex_perm, tells you how the vertices of the old triangulation correspond to the vertices of the new.
If return_edge_consequences, tells you what happened to the edges: if an edge e survived then edge_consequences[e.index()] gives you the new index, if not then it returns None."""
### perform = True isn't yet implemented for branch
### note if branch != None and this function does not return False, then there is only one possible branch
# if branch != None:
# assert has_non_sing_semiflow(tri, branch) ### we are checking this on the output of pachner moves so we don't need to check it here
edge = tri.edge(edge_num)
if edge.degree() != 3:
return False
tets = []
tet_nums = []
vertices = []
if angle != None:
non_pi_tet_num = None
for i in range(3):
embed = edge.embedding(i)
tets.append(embed.simplex())
tet_nums.append(tets[i].index())
vertices.append(embed.vertices())
if angle != None:
if not there_is_a_pi_here(angle, embed):
assert non_pi_tet_num == None
non_pi_tet_num = embed.simplex().index()
local_non_pi_tet_num = i
tet_nums.sort()
if len(set([tet.index() for tet in tets])) != 3:
return False ### tetrahedra must be distinct
if branch == None and not perform:
return True ### taut 3-2 move is always possible if the 3-2 move is.
if angle != None:
### record the "slope" of the pis on the non_pi_tet. This is a boolean
non_pi_tet_positive = unsorted_vert_pair_to_edge_pair[ ( vertices[local_non_pi_tet_num][0], vertices[local_non_pi_tet_num][2] ) ]
is_positive_slope = (angle[non_pi_tet_num] == non_pi_tet_positive)
### check we do the same as regina...
tri2 = regina.Triangulation3(tri) ## make a copy
tri2.pachner(tri2.edge(edge_num))
if return_vertex_perm:
vertex_representatives = []
for c in tri.vertices():
embed = c.embedding(0)
vertex_representatives.append((embed.simplex(), embed.face())) ### pair (tet, vert_num_of_that_tet)
### for testing:
vertex_degrees = [v.degree() for v in tri.vertices()]
## record the tetrahedra and gluings adjacent to the tets
gluings = []
for i in range(3):
tet_gluings = []
for j in range(2):
tet_gluings.append( [ tets[i].adjacentTetrahedron(vertices[i][j]), tets[i].adjacentGluing(vertices[i][j])] )
gluings.append(tet_gluings)
for i in range(3):
assert tets[i].adjacentTetrahedron(vertices[i][2]) == tets[(i+1)%3] ### The edge embeddings should be ordered this way...
if branch != None:
large_edges = [] ### record large edge info for the outer faces
for i in range(3):
this_tet_large_edges = []
for j in range(2):
this_tet_large_edges.append(large_edge_of_face( branch[tets[i].index()], vertices[i][j] ))
large_edges.append(this_tet_large_edges)
### add new tetrahedra
new_tets = []
for i in range(2):
new_tets.append(tri.newTetrahedron())
### glue across face
new_tets[0].join(3, new_tets[1], regina.Perm4(0,2,1,3))
### replace mapping info with corresponding info for the 2 tet. Self gluings will be annoying...
### write vertices[i][j] as vij
### tets[0] new_tet1
### _________ _________
### ,'\`.v00,'/`. ,'\ /`.
### ,' \ `.' / `. ,' \ 3 / `.
### ,' v10\ | /v20 `. ,' \ / `.
### /|\ \|/ /|\ / \ \ / / \
### / | \ * / | \ / \ * / \
### v12/ | \..... | ...../ | \v23 / 1 _\..... | ...../_ 2 \
### / ,' / * \ `. \ /_--"" / * \ ""--_\
### \`.| /v03 /|\ v02\ |,'/ `. \`. 2 / / \ \ 1 ,'/
### \ `./ / | \ \,' / ----} \ `./ / \ \,' /
### \|/`. / | \ ,'\|/ ,' \ /`. / 0 \ ,'\ /
### \ `. / | \ ,' / \ `. / \ ,' /
### \ * v13|v22 * / \ `---------' /
### \ \ | / / \ \ / /
### \ \ | / / \ \ 0 / /
### \ \ | / / \ \ / /
### tets[1] \ \|/ / tets[2] \ \ / /
### \ * / \ * / new_tet0
### \..v01../ \...|.../
### \`.|.'/ \ | /
### v11\ | /v21 \ 3 /
### \|/ \|/
### * *
# permutations taking the vertices for a face of the 2-tet ball to the
# vertices of the same face for the 3-tet ball
# these should be even in order to preserve orientability.
# perms = [[regina.Perm4( vertices[0][0], vertices[0][2], vertices[0][3], vertices[0][1] ), ### opposite v00
# regina.Perm4( vertices[0][1], vertices[0][3], vertices[0][2], vertices[0][0] ) ### opposite v01
# ],
# [regina.Perm4( vertices[1][3], vertices[1][0], vertices[1][2], vertices[1][1] ), ### opposite v10
# regina.Perm4( vertices[1][3], vertices[1][2], vertices[1][1], vertices[1][0] ) ### opposite v11
# ],
# [regina.Perm4( vertices[2][2], vertices[2][3], vertices[2][0], vertices[2][1] ), ### opposite v20
# regina.Perm4( vertices[2][2], vertices[2][1], vertices[2][3], vertices[2][0] ) ### opposite v21
# ]
# ]
perms = [[vertices[0] * regina.Perm4( 0, 2, 3, 1 ), ### opposite v00
vertices[0] * regina.Perm4( 1, 3, 2, 0 ) ### opposite v01
],
[vertices[1] * regina.Perm4( 3, 0, 2, 1 ), ### opposite v10
vertices[1] * regina.Perm4( 3, 2, 1, 0 ) ### opposite v11
],
[vertices[2] * regina.Perm4( 2, 3, 0, 1 ), ### opposite v20
vertices[2] * regina.Perm4( 2, 1, 3, 0 ) ### opposite v21
]
]
for i in range(3):
for j in range(2):
gluing = gluings[i][j]
if gluing != None:
if gluing[0] not in tets: ### not a self gluing
gluing[1] = gluing[1] * perms[i][j]
else:
i_other = tets.index( gluing[0] )
otherfacenum = gluing[1][vertices[i][j]]
j_other = [vertices[i_other][k] for k in range(4)].index(otherfacenum)
assert gluings[i_other][j_other][0] == tets[i]
assert gluings[i_other][j_other][1].inverse() == gluings[i][j][1]
gluings[i_other][j_other] = None ### only do a self gluing from one side
gluing[0] = new_tets[j_other] ### j refers to the vertex on the same 3 side
gluing[1] = perms[i_other][j_other].inverse() * gluing[1] * perms[i][j]
if return_vertex_perm:
new_vertex_representatives = []
for (tet, vert_num) in vertex_representatives:
if tet in tets:
which_tet = tets.index(tet)
vert_num = vertices[which_tet].inverse()[vert_num]
if vert_num == 0:
new_vertex_representatives.append((new_tets[1], 3))
elif vert_num == 1:
new_vertex_representatives.append((new_tets[0], 3))
elif vert_num == 2:
new_vertex_representatives.append((new_tets[0], (which_tet + 1) % 3 ))
elif vert_num == 3:
new_vertex_representatives.append((new_tets[0], (which_tet - 1) % 3 ))
else:
new_vertex_representatives.append((tet, vert_num)) ### not changed by the move
# ### for testing:
polar_cusp_indices = [ tets[0].vertex( vertices[0][0] ).index(), tets[0].vertex( vertices[0][1] ).index() ]
### unglue three tetrahedra
for tet in tets:
tet.isolate()
### remove the tetrahedra
for tet in tets:
tri.removeSimplex(tet)
### make the gluings on the boundary of the new ball
for i in range(3):
for j in range(2):
if gluings[i][j] != None:
if j == 0 or i == 0:
assert new_tets[j].adjacentTetrahedron(i) == None ## not glued
assert gluings[i][j][0].adjacentTetrahedron(gluings[i][j][1][i]) == None
new_tets[j].join(i, gluings[i][j][0], gluings[i][j][1])
else:
assert new_tets[j].adjacentTetrahedron(3 - i) == None ## not glued
assert gluings[i][j][0].adjacentTetrahedron(gluings[i][j][1][3 - i]) == None
new_tets[j].join(3 - i, gluings[i][j][0], gluings[i][j][1]) ## swap 1 and 2
assert tri.isIsomorphicTo(tri2)
assert tri.isOriented()
if angle != None:
### update the angle structure
angle.pop(tet_nums[2])
angle.pop(tet_nums[1])
angle.pop(tet_nums[0]) ## remove from the list in the correct order!
if local_non_pi_tet_num == 0:
if is_positive_slope:
new_angle = [0, 0]
else:
new_angle = [1, 1]
elif local_non_pi_tet_num == 1:
if is_positive_slope:
new_angle = [2, 1]
else:
new_angle = [0, 2]
else:
assert local_non_pi_tet_num == 2
if is_positive_slope:
new_angle = [1, 2]
else:
new_angle = [2, 0]
angle.extend(new_angle)
assert is_taut(tri, angle)
if branch != None:
### update the branched surface
branch.pop(tet_nums[2])
branch.pop(tet_nums[1])
branch.pop(tet_nums[0]) ## remove from the list in the correct order!
### for each of the two new tetrahedra, figure out what their outer face train tracks are
large_edges_new = [] ### record large edge info for the outer faces
for i in range(3):
this_tet_large_edges_new = []
for j in range(2):
new_large_edge = perms[i][j].inverse()[ large_edges[i][j] ]
this_tet_large_edges_new.append( new_large_edge )
large_edges_new.append(this_tet_large_edges_new)
large_edges_new_transposed = [list(i) for i in zip(*large_edges_new)]
branch0 = determine_possible_branch_given_three_faces([0,1,2], large_edges_new_transposed[0])
branch1 = determine_possible_branch_given_three_faces([0,2,1], large_edges_new_transposed[1])
if branch0 == None or branch1 == None:
return False
large_edge_for_new_tet0 = large_edge_of_face( branch0, 3 )
large_edge_for_new_tet1 = large_edge_of_face( branch1, 3 )
if large_edge_for_new_tet0 == large_edge_for_new_tet1:
if large_edge_for_new_tet0 != 0:
return False
else:
if large_edge_for_new_tet0 + large_edge_for_new_tet1 != 3: ### one must be 1, one must be 2
return False
branch.extend([branch0, branch1])
assert is_branched(tri, branch)
if not has_non_sing_semiflow(tri, branch):
return False
if return_vertex_perm:
vertex_permutation = []
for (tet, vert_num) in new_vertex_representatives:
vertex_permutation.append(tet.vertex(vert_num).index())
### note that Regina's permutations can have up to 16 entries, lets just use lists for this
### for testing:
new_vertex_degrees = [v.degree() for v in tri.vertices()]
new_vertex_degrees_pulled_back = [new_vertex_degrees[vertex_permutation[i]] for i in range(len(vertex_degrees))]
#if new_vertex_degrees != new_vertex_degrees_pulled_back:
# print(vertex_permutation)
# if vertex_permutation != list(range(len(new_vertex_representatives))):
# print('3-2 permuted vertices', vertex_permutation, new_vertex_degrees, new_vertex_degrees_pulled_back, vertex_degrees)
for i in polar_cusp_indices:
vertex_degrees[i] -= 2
assert vertex_degrees == new_vertex_degrees_pulled_back
# print(new_vertex_degrees_pulled_back)
output = [tri]
if angle != None:
output.append(angle)
if branch != None:
output.append(branch)
if return_triangle:
output.append(new_tets[0].triangle(3).index())
if return_vertex_perm:
output.append(vertex_permutation)
return output
def twoThreeMove(tri, face_num, angle = None, branch = None, perform = True, return_edge = False, return_vertex_perm = False, return_edge_consequences = False):
"""Apply a 2-3 move to a triangulation with a taut structure and/or branched surface, if possible.
If perform = False, returns if the move is possible.
If perform = True, modifies tri, returns (tri, angle, possible_branches) for the performed move
If return_edge, tells you the index of the newly created edge in the triangulation.
If return_vertex_perm, tells you how the vertices of the old triangulation correspond to the vertices of the new.
If return_edge_consequences, tells you what happened to the edges: edge_consequences[e.index()] gives you the new index."""
### possible_branches is a list
# if branch != None:
# assert has_non_sing_semiflow(tri, branch) ### we are checking this on the output of pachner moves so we don't need to check it here
## Joe Christy says [p764, Branched surfaces and attractors I: Dynamic Branched Surfaces] that if the branched surface carries the stable lamination of a pseudo-Anosov flow then it has a non singular semi flow
## We hope that we can move to a veering triangulation through such branched surfaces
face = tri.triangle(face_num)
embed0 = face.embedding(0)
tet0 = embed0.simplex()
tet_num0 = tet0.index()
tet_0_face_num = embed0.face()
vertices0 = embed0.vertices() # Maps vertices (0,1,2) of face to the corresponding vertex numbers of tet0
embed1 = face.embedding(1)
tet1 = embed1.simplex()
tet_num1 = tet1.index()
tet_1_face_num = embed1.face()
vertices1 = embed1.vertices() # Maps vertices (0,1,2) of face to the corresponding vertex numbers of tet1
if tet0 == tet1: ### Cannot perform a 2-3 move across a self-gluing
return False
if angle != None:
### taut 2-3 move is valid if the pis are on different edges of face
### this never happens if we start with a veering triangulation.
### for veering, the two-tetrahedron ball is always a continent.
for i in range(3):
j = (i+1) % 3
k = (i+2) % 3
if angle[tet_num0] == unsorted_vert_pair_to_edge_pair[(vertices0[j], vertices0[k])]:
pi_num_0 = i
if angle[tet_num1] == unsorted_vert_pair_to_edge_pair[(vertices1[j], vertices1[k])]:
pi_num_1 = i
if pi_num_0 == pi_num_1:
return False
if perform == False:
return True
### are all moves valid for the branched surface?
### for now, lets assume yes
### check we do the same as regina...
tri2 = regina.Triangulation3(tri) ## make a copy
tri2.pachner(tri2.triangle(face_num))
if return_vertex_perm:
vertex_representatives = []
for c in tri.vertices():
embed = c.embedding(0)
vertex_representatives.append((embed.simplex(), embed.face())) ### pair (tet, vert_num_of_that_tet)
### for testing:
vertex_degrees = [v.degree() for v in tri.vertices()]
if return_edge_consequences:
edge_representatives = []
for e in tri.edges():
embed = e.embedding(0)
### HERE
##edge_representatives.append((embed.simplex(), embed.face())) ### pair (tet, vert_num_of_that_tet)
### We have to implement twoThreeMove ourselves. e.g. we do a 2-3 move to canonical fig 8 knot complement triangulation.
### All of the original tetrahedra are removed. I don't see any way to carry the angle structure through without knowing
### exactly how Ben's implementation works.
## record the tetrahedra and gluings adjacent to tet0 and tet1
tets = [tet0, tet1]
vertices = [vertices0, vertices1]
gluings = []
for i in range(2):
tet_gluings = []
for j in range(3):
tet_gluings.append( [ tets[i].adjacentTetrahedron(vertices[i][j]), tets[i].adjacentGluing(vertices[i][j])] )
# if tets[i].adjacentTetrahedron(vertices[i][j]) in tets:
# print('self gluing')
gluings.append(tet_gluings)
if branch != None:
large_edges = [] ### record large edge info for the outer faces
for i in range(2):
this_tet_large_edges = []
for j in range(3):
this_tet_large_edges.append(large_edge_of_face( branch[tets[i].index()], vertices[i][j] ))
large_edges.append(this_tet_large_edges)
### add new tetrahedra
new_tets = []
for i in range(3):
new_tets.append(tri.newTetrahedron())
### glue around degree 3 edge
for i in range(3):
new_tets[i].join(2, new_tets[(i+1)%3], regina.Perm4(0,1,3,2))
### replace mapping info with corresponding info for the 3 tet. Self gluings will be annoying...
### write verticesi[j] as vij
### tet0 new_tets[0]
### _________ _________
### ,'\ /`. ,'\`. ,'/`.
### ,' \ v03 / `. ,' \ `0' / `.
### ,' \ / `. ,' \ | / `.
### / \ \ / / \ /|\ \|/ /|\
### /v02\ * /v01\ / | \ * / | \
### / _\..... | ...../_ \ / | 3\..... | ...../2 | \
### /_--"" / * \ ""--_\ /2 ,' / * \ `. 3\
### \`.v12/ / \ \v11,'/ `. \`.| / /|\ \ |,'/
### \ `./ / \ \,' / ----} \ `./ / | \ \,' /
### \ /`. / v00 \ ,'\ / ,' \|/`. / | \ ,'\|/
### \ `. / \ ,' / \ `. / | \ ,' /
### \ `---------' / \ * 3 | 2 * /
### \ \ / / \ \ | / /
### \ \ v10 / / new_tets[1] \ \ | / / new_tets[2]
### \ \ / / \ \ | / /
### \ \ / / \ \|/ /
### \ * / \ * /
### tet1 \...|.../ \...|.../
### \ | / \`.|.'/
### \v13/ \ 1 /
### \|/ \|/
### * *
# permutations taking the vertices for a face of the 3-tet ball to the
# vertices of the same face for the 2-tet ball
# these should be even in order to preserve orientability.
# exactly one of vertices[0] and vertices[1] is even, but it seems to depend on the face.
perms = [[vertices[0] * regina.Perm4( 3,0,1,2 ), ### opposite v00
vertices[0] * regina.Perm4( 3,1,2,0 ), ### opposite v01
vertices[0] * regina.Perm4( 3,2,0,1 ) ### opposite v02
],
[vertices[1] * regina.Perm4( 0,3,1,2 ), ### opposite v10
vertices[1] * regina.Perm4( 1,3,2,0 ), ### opposite v11
vertices[1] * regina.Perm4( 2,3,0,1 ) ### opposite v12
]
]
flip = perms[0][0].sign() == -1
if flip: #then all of the signs are wrong, switch 0 and 1 on input
perms = [[p * regina.Perm4( 1,0,2,3 ) for p in a] for a in perms]
for i in range(2):
for j in range(3):
gluing = gluings[i][j]
if gluing != None:
if gluing[0] not in tets: ### not a self gluing
gluing[1] = gluing[1] * perms[i][j]
else:
i_other = tets.index( gluing[0] )
otherfacenum = gluing[1][vertices[i][j]]
j_other = [vertices[i_other][k] for k in range(4)].index(otherfacenum)
assert gluings[i_other][j_other][0] == tets[i]
assert gluings[i_other][j_other][1].inverse() == gluings[i][j][1]
gluings[i_other][j_other] = None ### only do a self gluing from one side
gluing[0] = new_tets[j_other]
gluing[1] = perms[i_other][j_other].inverse() * gluing[1] * perms[i][j]
if return_vertex_perm:
new_vertex_representatives = []
for (tet, vert_num) in vertex_representatives:
if tet == tet0:
triangle_vert_num = vertices[0].inverse()[vert_num]
if triangle_vert_num == 3:
if flip:
new_vertex_representatives.append((new_tets[0], 1))
else:
new_vertex_representatives.append((new_tets[0], 0))
else:
new_vertex_representatives.append((new_tets[(triangle_vert_num + 1) % 3], 3))
elif tet == tet1:
triangle_vert_num = vertices[1].inverse()[vert_num]
if triangle_vert_num == 3:
if flip:
new_vertex_representatives.append((new_tets[0], 0))
else:
new_vertex_representatives.append((new_tets[0], 1))
else:
new_vertex_representatives.append((new_tets[(triangle_vert_num + 1) % 3], 3))
else:
new_vertex_representatives.append((tet, vert_num)) ### not changed by the move
### for testing:
polar_cusp_indices = [ tet0.vertex( vertices[0][3] ).index(), tet1.vertex( vertices[1][3] ).index() ]
### unglue two tetrahedra
tet0.isolate()
tet1.isolate()
### remove the tetrahedra
tri.removeSimplex(tet0)
tri.removeSimplex(tet1)
### make the gluings on the boundary of the new ball
for i in range(2):
for j in range(3):
if gluings[i][j] != None:
if flip:
new_tets[j].join(i, gluings[i][j][0], gluings[i][j][1])
else:
new_tets[j].join(1 - i, gluings[i][j][0], gluings[i][j][1])
assert tri.isIsomorphicTo(tri2)
assert tri.isOriented()
if angle != None:
### update the angle structure
tet_indices = [tet_num0, tet_num1]
tet_indices.sort()
angle.pop(tet_indices[1])
angle.pop(tet_indices[0]) ## remove from the list in the correct order!
new_angle = [None, None, None]
new_angle[pi_num_0] = 0
new_angle[pi_num_1] = 0 ### these two tetrahedra have their pi's on the new degree three edge
third_index = 3 - (pi_num_0 + pi_num_1)
if (pi_num_0 - third_index) % 3 == 1:
new_angle[third_index] = 1
else:
assert (pi_num_0 - third_index) % 3 == 2
new_angle[third_index] = 2
if flip:
new_angle[third_index] = 3 - new_angle[third_index]
angle.extend(new_angle)
assert is_taut(tri, angle)
if branch != None:
### update the branched surface
### for each of the three new tetrahedra, figure out what their outer face train tracks are
large_edges_new = [] ### record large edge info for the outer faces
for i in range(2):
this_tet_large_edges_new = []
for j in range(3):
new_large_edge = perms[i][j].inverse()[ large_edges[i][j] ]
if flip:
assert new_large_edge != i ### the face number cannot be the large vertex for that face
else:
assert new_large_edge != 1 - i
this_tet_large_edges_new.append( new_large_edge )
large_edges_new.append(this_tet_large_edges_new)
candidate_branches = []
for j in range(3):
if flip:
candidate_branches.append( determine_possible_branch_given_two_faces((0,1), (large_edges_new[0][j], large_edges_new[1][j]) ) )
else:
candidate_branches.append( determine_possible_branch_given_two_faces((1,0), (large_edges_new[0][j], large_edges_new[1][j]) ) )
### update the branch structure, many possible ways
tet_indices = [tet_num0, tet_num1]
tet_indices.sort()
branch.pop(tet_indices[1])
branch.pop(tet_indices[0]) ## remove from the list in the correct order!
out_branches = []
for cand0 in candidate_branches[0]:
for cand1 in candidate_branches[1]:
for cand2 in candidate_branches[2]:
candidate = branch[:] + [cand0, cand1, cand2]
# print('candidate', candidate)
# if is_branched(tri, candidate):
if has_non_sing_semiflow(tri, candidate):
out_branches.append(candidate)
# assert len(out) > 0 ### this works if we check is_branched three lines above, but not if we check has_non_sing_semiflow
if len(out_branches) == 0: ### with has_non_sing_semiflow instead, we might not get any
return False
if return_vertex_perm:
vertex_permutation = []
for (tet, vert_num) in new_vertex_representatives:
vertex_permutation.append(tet.vertex(vert_num).index())
### note that Regina's permutations can have up to 16 entries, lets just use lists for this
### for testing:
new_vertex_degrees = [v.degree() for v in tri.vertices()]
new_vertex_degrees_pulled_back = [new_vertex_degrees[vertex_permutation[i]] for i in range(len(vertex_degrees))]
# print(vertex_permutation)
# if vertex_permutation != list(range(len(new_vertex_representatives))):
# print('2-3 permuted vertices', vertex_permutation, new_vertex_degrees, new_vertex_degrees_pulled_back, vertex_degrees)
for i in polar_cusp_indices:
vertex_degrees[i] += 2
assert vertex_degrees == new_vertex_degrees_pulled_back
# print(new_vertex_degrees_pulled_back)
output = [tri]
if angle != None:
output.append(angle)
if branch != None:
output.append(out_branches)
if return_edge:
output.append(new_tets[0].edge(0).index())
if return_vertex_perm:
output.append(vertex_permutation)
return output
def run_tests(num_to_check=10, smaller_num_to_check = 10):
import taut
veering_isosigs = parse_data_file("Data/veering_census.txt")
print("testing is_taut")
for sig in random.sample(veering_isosigs, num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
assert taut.is_taut(tri, angle), sig
print("testing isosig round trip")
for sig in random.sample(veering_isosigs, num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
recovered_sig = taut.isosig_from_tri_angle(tri, angle)
assert sig == recovered_sig, sig
# we only test this round trip - the other round trip does not
# make sense because tri->isosig is many to one.
import transverse_taut
print("testing is_transverse_taut")
for sig in random.sample(veering_isosigs, num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
assert transverse_taut.is_transverse_taut(tri, angle), sig
non_transverse_taut_isosigs = parse_data_file("Data/veering_non_transverse_taut_examples.txt")
print("testing not is_transverse_taut")
for sig in non_transverse_taut_isosigs:
tri, angle = taut.isosig_to_tri_angle(sig)
assert not transverse_taut.is_transverse_taut(tri, angle), sig
import veering
print("testing is_veering")
for sig in random.sample(veering_isosigs, num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
assert veering.is_veering(tri, angle), sig
# tri, angle = taut.isosig_to_tri_angle("cPcbbbdxm_10")
# explore_mobius_surgery_graph(tri, angle, max_tetrahedra = 12)
# # tests to see that it makes only veering triangulations as it goes
import veering_dehn_surgery
print("testing veering_dehn_surgery")
for sig in random.sample(veering_isosigs, num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
for face_num in veering_dehn_surgery.get_mobius_strip_indices(tri):
(tri_s, angle_s, face_num_s) = veering_dehn_surgery.veering_mobius_dehn_surgery(tri, angle, face_num)
assert veering.is_veering(tri_s, angle_s), sig
import veering_fan_excision
print("testing veering_fan_excision")
m003, _ = taut.isosig_to_tri_angle('cPcbbbdxm_10')
m004, _ = taut.isosig_to_tri_angle('cPcbbbiht_12')
for sig in random.sample(veering_isosigs, num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
tet_types = veering.is_veering(tri, angle, return_type = "tet_types")
if tet_types.count("toggle") == 2:
excised_tri, _ = veering_fan_excision.excise_fans(tri, angle)
assert ( excised_tri.isIsomorphicTo(m003) != None or
excised_tri.isIsomorphicTo(m004) != None ), sig
import pachner
print("testing pachner with taut structure")
for sig in random.sample(veering_isosigs, num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
face_num = random.randrange(tri.countTriangles())
result = pachner.twoThreeMove(tri, face_num, angle = angle, return_edge = True)
if result != False:
tri2, angle2, edge_num = result
tri3, angle3 = pachner.threeTwoMove(tri2, edge_num, angle = angle2)
assert taut.isosig_from_tri_angle(tri, angle) == taut.isosig_from_tri_angle(tri3, angle3), sig
import branched_surface
import regina
print("testing branched_surface and pachner with branched surface")
for sig in random.sample(veering_isosigs, num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
tri_original = regina.Triangulation3(tri) #copy
branch = branched_surface.upper_branched_surface(tri, angle, return_lower = random.choice([True, False]))
### test branch isosig round trip
sig_with_branch = branched_surface.isosig_from_tri_angle_branch(tri, angle, branch)
tri2, angle2, branch2 = branched_surface.isosig_to_tri_angle_branch(sig_with_branch)
assert (branch == branch2) and (angle == angle2), sig
branch_original = branch[:] #copy
face_num = random.randrange(tri.countTriangles())
out = pachner.twoThreeMove(tri, face_num, branch = branch, return_edge = True)
if out != False:
tri, possible_branches, edge_num = out
tri, branch = pachner.threeTwoMove(tri, edge_num, branch = possible_branches[0])
all_isoms = tri.findAllIsomorphisms(tri_original)
all_branches = [branched_surface.apply_isom_to_branched_surface(branch, isom) for isom in all_isoms]
assert branch_original in all_branches, sig
import flow_cycles
import drill
print("testing taut and branched drill + semiflows on drillings")
for sig in random.sample(veering_isosigs, smaller_num_to_check):
tri, angle = taut.isosig_to_tri_angle(sig)
branch = branched_surface.upper_branched_surface(tri, angle) ### also checks for veering and transverse taut
found_loops = flow_cycles.find_flow_cycles(tri, branch)
for loop in random.sample(found_loops, min(len(found_loops), 5)): ## drill along at most 5 loops
tri, angle = taut.isosig_to_tri_angle(sig)
branch = branched_surface.upper_branched_surface(tri, angle)
tri_loop = flow_cycles.flow_cycle_to_triangle_loop(tri, branch, loop)
if tri_loop != False:
if not flow_cycles.tri_loop_is_boundary_parallel(tri_loop, tri):
drill.drill(tri, tri_loop, angle = angle, branch = branch, sig = sig)
assert branched_surface.has_non_sing_semiflow(tri, branch), sig
print("all basic tests passed")
try:
import snappy
import snappy_util
snappy_working = True
except:
print("failed to import from snappy?")
snappy_working = False
if snappy_working:
print("testing algebraic intersection")
census = snappy.OrientableCuspedCensus() # not a set or list, so can't use random.sample
for i in range(10):
M = random.choice(census)
n = M.num_cusps()
peripheral_curves = M.gluing_equations()[-2*n:]
for i in range(2*n):
for j in range(i, 2*n):
alg_int = snappy_util.algebraic_intersection(peripheral_curves[i], peripheral_curves[j])
if i % 2 == 0 and j == i + 1:
assert alg_int == 1, M.name()
else:
assert alg_int == 0, M.name()
if snappy_working:
import veering_drill_midsurface_bdy
print("testing veering drilling and filling")
for sig in random.sample(veering_isosigs[:3000], num_to_check):
T, per = veering_drill_midsurface_bdy.drill_midsurface_bdy(sig)
M = snappy.Manifold(T.snapPea())
M.set_peripheral_curves("shortest")
L = snappy_util.get_slopes_from_peripherals(M, per)
M.dehn_fill(L)
N = snappy.Manifold(sig.split("_")[0])
assert M.is_isometric_to(N), sig
if snappy_working:
print("all tests depending on snappy passed")
# try:
# from hashlib import md5
# from os import remove
# import pyx
# from boundary_triangulation import draw_triangulation_boundary_from_veering_isosig
# pyx_working = True
# except:
# print("failed to import from pyx?")
# pyx_working = False
# ladders_style_sigs = {
# "cPcbbbiht_12": "f34c1fdf65db9d02994752814803ae01",
# "gLLAQbecdfffhhnkqnc_120012": "091c85b4f4877276bfd8a955b769b496",
# "kLALPPzkcbbegfhgijjhhrwaaxnxxn_1221100101": "a0f15a8454f715f492c74ce1073a13a4",
# }
# geometric_style_sigs = {
# "cPcbbbiht_12": "1e74d0b68160c4922e85a5adb20a0f1d",
# "gLLAQbecdfffhhnkqnc_120012": "856a1fce74eb64f519bcda083303bd8f",
# "kLALPPzkcbbegfhgijjhhrwaaxnxxn_1221100101": "33bd23b34c5d977a103fa50ffe63120a",
# }
# args = {
# "draw_boundary_triangulation":True,
# "draw_triangles_near_poles": False,
# "ct_depth":-1,
# "ct_epsilon":0.03,
# "global_drawing_scale": 4,
# "delta": 0.2,
# "ladder_width": 10.0,
# "ladder_height": 20.0,
# "draw_labels": True,
# }
# shapes_data = read_from_pickle("Data/veering_shapes_up_to_ten_tetrahedra.pkl")
# if pyx_working:
# for sig in ladders_style_sigs:
# print("testing boundary triangulation pictures, ladder style", sig)
# args["tet_shapes"] = shapes_data[sig]
# args["style"] = "ladders"
# file_name = draw_triangulation_boundary_from_veering_isosig(sig, args = args)
# f = open(file_name, "rb")
# file_hash = md5(f.read())
# assert file_hash.hexdigest() == ladders_style_sigs[sig]
# f.close()
# remove(file_name)
# if pyx_working:
# for sig in geometric_style_sigs:
# print("testing boundary triangulation pictures, ladder style", sig)
# args["tet_shapes"] = shapes_data[sig]
# args["style"] = "geometric"
# file_name = draw_triangulation_boundary_from_veering_isosig(sig, args = args)
# f = open(file_name, "rb")
# file_hash = md5(f.read())
# assert file_hash.hexdigest() == geometric_style_sigs[sig]
# f.close()
# remove(file_name)
# if pyx_working:
# print("all tests depending on pyx passed")
veering_polys = {
"cPcbbbiht_12": [-4, -1, 1, 4],
"eLMkbcddddedde_2100": [-2, -2, -2, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 2, 2],
"gLLAQbecdfffhhnkqnc_120012": [-1, -1, -1, -1, 1, 1, 1, 1],
"gLLPQcdfefefuoaaauo_022110": [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1],
}
# veering_polys = { ### old
# "cPcbbbiht_12": "a^3 - 4*a^2 + 4*a - 1",
# "eLMkbcddddedde_2100": "a^6*b - a^6 - 2*a^5*b - a^4*b^2 + a^5 + 2*a^4*b + a^3*b^2 - 2*a^3*b + a^3 + 2*a^2*b + a*b^2 - a^2 - 2*a*b - b^2 + b",
# "gLLAQbecdfffhhnkqnc_120012": "a^7 + a^6 + a^5 + a^4 - a^3 - a^2 - a - 1",
# "gLLPQcdfefefuoaaauo_022110": "a^12*b^3 - a^11*b^2 - a^10*b^3 - a^10*b^2 - a^7*b^3 - a^7*b^2 - a^6*b^3 + a^7*b + a^5*b^2 - a^6 - a^5*b - a^5 - a^2*b - a^2 - a*b + 1",
# }
taut_polys = {
"cPcbbbiht_12": [-3, 1, 1],
"eLMkbcddddedde_2100": [-1, -1, -1, 1, 1],
"iLLAwQcccedfghhhlnhcqeesr_12001122": [],
}
# taut_polys = { ### old
# "cPcbbbiht_12": "a^2 - 3*a + 1",
# "eLMkbcddddedde_2100": "a^2*b - a^2 - a*b - b^2 + b",
# "iLLAwQcccedfghhhlnhcqeesr_12001122": "0",
# }
torus_bundles = [
"cPcbbbiht_12",
"eLMkbcdddhhqqa_1220",
"gLMzQbcdefffhhqqqdl_122002",
]
measured = [
"gLLAQbecdfffhhnkqnc_120012",
"iLLALQcccedhgghhlnxkxrkaa_12001112",
"iLLAwQcccedfghhhlnhcqeesr_12001122",
]
empties = [
"fLAMcaccdeejsnaxk_20010",
"gLALQbcbeeffhhwsras_211220",
"hLALAkbcbeefgghhwsraqj_2112202",
]
try:
from sage.rings.integer_ring import ZZ
sage_working = True
except:
print("failed to import from sage?")
sage_working = False
if sage_working:
import taut_polytope
print("testing is_layered")
for sig in veering_isosigs[:17]:
assert taut_polytope.is_layered(sig), sig
for sig in veering_isosigs[17:21]:
assert not taut_polytope.is_layered(sig), sig
if sage_working:
import fibered
print("testing is_fibered")
mflds = parse_data_file("Data/mflds_which_fiber.txt")
mflds = [line.split("\t")[0:2] for line in mflds]
for (name, kind) in random.sample(mflds, num_to_check):
assert fibered.is_fibered(name) == (kind == "fibered"), name
if sage_working:
import veering_polynomial
import taut_polynomial
print("testing veering poly")
for sig in veering_polys:
p = veering_polynomial.veering_polynomial(sig)
assert check_polynomial_coefficients(p, veering_polys[sig]), sig
### Nov 2021: sage 9.4 changed how smith normal form works, which changed our polynomials
### to equivalent but not equal polynomials. To avoid this kind of change breaking things
### in the future, we changed to comparing the list of coefficients.
# assert p.__repr__() == veering_polys[sig]
print("testing taut poly")
for sig in taut_polys:
p = taut_polynomial.taut_polynomial_via_tree(sig)
assert check_polynomial_coefficients(p, taut_polys[sig]), sig
# assert p.__repr__() == taut_polys[sig]
print("testing divide")
for sig in random.sample(veering_isosigs[:3000], num_to_check):
p = veering_polynomial.veering_polynomial(sig)
q = taut_polynomial.taut_polynomial_via_tree(sig)
if q == 0:
assert p == 0, sig
else:
assert q.divides(p), sig
if sage_working:
print("testing alex")
for sig in random.sample(veering_isosigs[:3000], num_to_check):
snap_sig = sig.split("_")[0]
M = snappy.Manifold(snap_sig)
if M.homology().betti_number() == 1:
assert taut_polynomial.taut_polynomial_via_tree(sig, mode = "alexander") == M.alexander_polynomial(), sig
if sage_working:
# would be nice to automate this - need to fetch the angle
# structure say via z_charge.py...
print("testing is_torus_bundle")
for sig in torus_bundles:
assert taut_polytope.is_torus_bundle(sig), sig
if sage_working:
# ditto
print("testing is_layered")
for sig in torus_bundles:
assert taut_polytope.is_layered(sig), sig
print("testing measured")
for sig in measured:
assert taut_polytope.LMN_tri_angle(sig) == "M", sig
print("testing empty")
for sig in empties:
assert taut_polytope.LMN_tri_angle(sig) == "N", sig
if sage_working: # warning - this takes random amounts of time!
print("testing hom dim")
for sig in random.sample(veering_isosigs[:3000], 3): # magic number
# dimension = zero if and only if nothing is carried.
assert (taut_polytope.taut_cone_homological_dim(sig) == 0) == (taut_polytope.LMN_tri_angle(sig) == "N"), sig
if sage_working:
boundary_cycles = {
("eLMkbcddddedde_2100",(2,5,5,1,3,4,7,1)): "((-7, -7, 0, 0, 4, -3, 7, 0), (7, 7, 0, 0, -4, 3, -7, 0))",
("iLLLQPcbeegefhhhhhhahahha_01110221",(0,1,0,0,0,1,0,0,0,0,0,0,1,0,1,0)): "((0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, 0, 0, 1, 0, 1), (0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0, 0, 0, -1, 0, -1))",
("ivvPQQcfhghgfghfaaaaaaaaa_01122000",(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)): "((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2), (1, 1, 0, 2, -1, 0, -3, 1, 2, -1, -2, 0, 3, -2, -1, 0), (-2, 0, -3, 1, 2, -1, 0, 2, -1, 0, 3, 1, -2, 1, 0, -1), (0, -2, 1, -3, 0, -1, 2, 0, -1, 2, -1, 1, 0, 1, -2, 3))",
}
taut_polys_with_cycles = {
("eLMkbcddddedde_2100", ((7, 7, 0, 0, -4, 3, -7, 0),)): [-1, -1, -1, 1, 1],
("iLLLQPcbeegefhhhhhhahahha_01110221", ((0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0, 0, 0, -1, 0, -1),)): [1, 1, 2],
("ivvPQQcfhghgfghfaaaaaaaaa_01122000", ((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2), (1, 1, 0, 2, -1, 0, -3, 1, 2, -1, -2, 0, 3, -2, -1, 0))): [-4, -1, -1, 1, 1],
}
# taut_polys_with_cycles = {
# ("eLMkbcddddedde_2100", ((7, 7, 0, 0, -4, 3, -7, 0),)): "a^14 - a^8 - a^7 - a^6 + 1",
# ("iLLLQPcbeegefhhhhhhahahha_01110221", ((0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0, 0, 0, -1, 0, -1),)): "a^2 + 2*a + 1",
# ("ivvPQQcfhghgfghfaaaaaaaaa_01122000", ((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2), (1, 1, 0, 2, -1, 0, -3, 1, 2, -1, -2, 0, 3, -2, -1, 0))): "a*b^2 - a^2 - 4*a*b - b^2 + a",
# }
taut_polys_image = {
('eLMkbcddddedde_2100', ((7, 8, -1, 0, -4, 4, -8, 0),)):[-1, -1, -1, 1, 1],
('ivvPQQcfhghgfghfaaaaaaaaa_01122000', ((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2),)):[-2, -2, -1, -1, 1, 1],
('ivvPQQcfhghgfghfaaaaaaaaa_01122000', ((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2), (1, 1, 0, 2, -1, 0, -3, 1, 2, -1, -2, 0, 3, -2, -1, 0))):[-4, -1, -1, 1, 1]
}
# taut_polys_image = {
# ('eLMkbcddddedde_2100', ((7, 8, -1, 0, -4, 4, -8, 0),)):"a^16 - a^9 - a^8 - a^7 + 1",
# ('ivvPQQcfhghgfghfaaaaaaaaa_01122000', ((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2),)):"a*b^2*c - 2*a*b*c - b^2*c - a^2 - 2*a*b + a",
# ('ivvPQQcfhghgfghfaaaaaaaaa_01122000', ((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2), (1, 1, 0, 2, -1, 0, -3, 1, 2, -1, -2, 0, 3, -2, -1, 0))):"a*b^2 - a^2 - 4*a*b - b^2 + a"
# }
alex_polys_with_cycles = {
("eLMkbcddddedde_2100",((7, 7, 0, 0, -4, 3, -7, 0),)): [-2, -1, -1, -1, 1, 1, 1, 2],
("iLLLQPcbeegefhhhhhhahahha_01110221", ((0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0, 0, 0, -1, 0, -1),)): [-3, -1, 1, 3],
("ivvPQQcfhghgfghfaaaaaaaaa_01122000", ((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2), (1, 1, 0, 2, -1, 0, -3, 1, 2, -1, -2, 0, 3, -2, -1, 0))): [-1, -1, 1, 1],
}
# alex_polys_with_cycles = {
# ("eLMkbcddddedde_2100",((7, 7, 0, 0, -4, 3, -7, 0),)): "a^15 - a^14 + a^9 - 2*a^8 + 2*a^7 - a^6 + a - 1",
# ("iLLLQPcbeegefhhhhhhahahha_01110221", ((0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0, 0, 0, -1, 0, -1),)): "3*a^3 - a^2 + a - 3",
# ("ivvPQQcfhghgfghfaaaaaaaaa_01122000", ((1, 1, 2, 0, -1, 2, 1, -3, 0, -1, 0, -2, -1, 0, 3, -2), (1, 1, 0, 2, -1, 0, -3, 1, 2, -1, -2, 0, 3, -2, -1, 0))): "a*b^2 - a^2 - b^2 + a",
# }
if sage_working:
import taut_carried
print("testing boundary cycles")
for sig, surface in boundary_cycles:
surface_list = list(surface)
cycles = taut_carried.boundary_cycles_from_surface(sig, surface_list)
cycles = tuple(tuple(cycle) for cycle in cycles)
assert cycles.__repr__() == boundary_cycles[(sig, surface)], sig
if sage_working:
print("testing taut with cycles")
for sig, cycles in taut_polys_with_cycles:
cycles_in = [list(cycle) for cycle in cycles]
p = taut_polynomial.taut_polynomial_via_tree(sig, cycles_in)
assert check_polynomial_coefficients(p, taut_polys_with_cycles[(sig, cycles)]), sig
# assert p.__repr__() == taut_polys_with_cycles[(sig, cycles)]
if sage_working:
print("testing taut with images")
for sig, cycles in taut_polys_image:
cycles_in = [list(cycle) for cycle in cycles]
p = taut_polynomial.taut_polynomial_image(sig, cycles_in)
assert check_polynomial_coefficients(p, taut_polys_image[(sig, cycles)]), sig
# assert p.__repr__() == taut_polys_image[(sig, cycles)]
if sage_working:
print("testing alex with cycles")
for sig, cycles in alex_polys_with_cycles:
cycles_in = [list(cycle) for cycle in cycles]
p = taut_polynomial.taut_polynomial_via_tree(sig, cycles_in, mode = "alexander")
assert check_polynomial_coefficients(p, alex_polys_with_cycles[(sig, cycles)]), sig
# assert p.__repr__() == alex_polys_with_cycles[(sig, cycles)]
if sage_working:
import edge_orientability
import taut_euler_class
print("testing euler and edge orientability")
for sig in random.sample(veering_isosigs[:3000], 3):
# Theorem: If (tri, angle) is edge orientable then e = 0.
assert not ( edge_orientability.is_edge_orientable(sig) and
(taut_euler_class.order_of_euler_class_wrapper(sig) == 2) ), sig
if sage_working:
# Theorem: If (tri, angle) is edge orientable then taut poly = alex poly.
# taut_polynomial.taut_polynomial_via_tree(sig, mode = "alexander") ==
# taut_polynomial.taut_polynomial_via_tree(sig, mode = "taut")
pass
if sage_working:
print("testing exotics")
for sig in random.sample(veering_isosigs[:3000], 3):
tri, angle = taut.isosig_to_tri_angle(sig)
T = veering.veering_triangulation(tri, angle)
is_eo = T.is_edge_orientable()
for angle in T.exotic_angles():
assert taut_polytope.taut_cone_homological_dim(tri, angle) == 0, sig
assert is_eo == transverse_taut.is_transverse_taut(tri, angle), sig
### test for drill_midsurface_bdy: drill then fill, check you get the same manifold
if sage_working:
from sage.combinat.words.word_generators import words
from sage.modules.free_module_integer import IntegerLattice
from sage.modules.free_module import VectorSpace
from sage.matrix.constructor import Matrix
import z_charge
import z2_taut
import regina
ZZ2 = ZZ.quotient(ZZ(2))
sig_starts = ["b+-LR", "b++LR"]
print("testing lattice for punc torus bundle")
for i in range(3):
for sig_start in sig_starts:
sig = sig_start + str(words.RandomWord(8, 2, "LR")) # 8 is a magic number
M = snappy.Manifold(sig)
tri = regina.Triangulation3(M)
t, A = z_charge.sol_and_kernel(M)
B = z_charge.leading_trailing_deformations(M)
C = z2_taut.cohomology_loops(tri)
AA = IntegerLattice(A)
BB = IntegerLattice(B)
assert AA == BB.saturation(), sig
dim = 3*M.num_tetrahedra()
V = VectorSpace(ZZ2, dim)
AA = V.subspace(A)
BB = V.subspace(B)
CM = Matrix(ZZ2, C)
CC = CM.right_kernel()
assert AA.intersection(CC) == BB , sig
## so l-t defms are the part of the kernel that doesn't flip over
if sage_working:
print("testing charges for punc torus bundle")
for i in range(3):
for sig_start in sig_starts:
sig = sig_start + str(words.RandomWord(8, 2, "LR")) # 8 is a magic number
M = snappy.Manifold(sig)
assert z_charge.can_deal_with_reduced_angles(M), sig
if sage_working:
import carried_surface
import mutation
print("testing building carried surfaces and mutations")
sigs_weights = [
['iLLLPQccdgefhhghqrqqssvof_02221000', (0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0)],
['jLLAvQQcedehihiihiinasmkutn_011220000', (2, 0, 1, 0, 0, 0, 1, 2, 0, 2, 0, 2, 1, 0, 0, 0, 1, 0)],
['jLLAvQQcedehihiihiinasmkutn_011220000', (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)],
['jLLLMPQcdgfhfhiiihshassspiq_122201101', (0, 0, 4, 0, 4, 1, 0, 2, 2, 0, 1, 0, 0, 4, 0, 4, 0, 0)]
]
strata = [
((1, 2), [2, 2]),
((2, 4), [5, 5, 1, 1]),
((0, 3), [2, 0, 0]),
((6, 1), [22])
]
orders_of_veering_symmetry_groups = [4, 2, 2, 2]
for i in range(len(sigs_weights)):
tri, angle = taut.isosig_to_tri_angle(sigs_weights[i][0])
weights = sigs_weights[i][1]
surface, edge_colours = carried_surface.build_surface(tri, angle, weights, return_edge_colours = True)
assert strata[i] == carried_surface.stratum_from_weights_surface(weights, surface)
veering_isoms = carried_surface.veering_symmetry_group(surface, edge_colours)
assert len(veering_isoms) == orders_of_veering_symmetry_groups[i]
isom = veering_isoms[1]
mutation.mutate(tri, angle, weights, isom, quiet = True)
if i == 0:
assert tri.isoSig() == 'ivLLQQccfhfeghghwadiwadrv'
#print('svof to wadrv passed')
elif i == 1:
assert tri.isoSig() == 'jvLLAQQdfghhfgiiijttmtltrcr'
#print('smkutn to tltrcr passed')
elif i == 2:
assert tri.isoSig() == 'jLLMvQQcedehhiiihiikiwnmtxk'
#print('smkutn to mtxk passed')
elif i == 3:
assert tri.isoSig() == 'jLLALMQcecdhggiiihqrwqwrafo'
#print('spiq to rafo passed')
if sage_working:
print("all tests depending on sage passed")
def is_veering(tri, angle, return_type="boolean"):
"""
checks to see if this triangulation with taut angle structure is
veering
"""
# return type can be "boolean", "veering_colours", "veering_partition" or "num_toggles"
# all return False if it is not veering. If it is veering:
# "veering_colours" returns the data of which edges are "blue" (bLue - Left veering) or "red" (Right veering) veering,
# "veering_partition" rewrites this as a list of the "blue" edges followed by a list of the "red" edges
# "num_toggles" returns the number of toggle tetrahedra
# "tet_types" returns list of tets, either "toggle", "red" or "blue" veering
assert is_taut(tri, angle)
# first do quick test on edge degrees
for e in tri.edges():
if e.degree() < 4:
return False
# now do proper check
veering_colours = ["-"] * tri.countEdges()
for tet_num in range(tri.countTetrahedra()):
tet = tri.tetrahedron(tet_num)
edgepair = angle[tet_num]
# now label edges L or R
L_edgepair = (
edgepair + 1
) % 3 # L or R depends on orientation, this takes us from pi edge pair to L edge pair
L_edge_1, L_edge_2 = L_edgepair, 5 - L_edgepair
L_edge_1_index = tet.edge(L_edge_1).index()
L_edge_2_index = tet.edge(L_edge_2).index()
if veering_colours[L_edge_1_index] == "red" or veering_colours[
L_edge_2_index] == "red":
return False
else:
veering_colours[L_edge_1_index] = "blue"
veering_colours[L_edge_2_index] = "blue"
R_edgepair = (edgepair + 2) % 3
R_edge_1, R_edge_2 = R_edgepair, 5 - R_edgepair
R_edge_1_index = tet.edge(R_edge_1).index()
R_edge_2_index = tet.edge(R_edge_2).index()
if veering_colours[R_edge_1_index] == "blue" or veering_colours[
R_edge_2_index] == "blue":
return False
else:
veering_colours[R_edge_1_index] = "red"
veering_colours[R_edge_2_index] = "red"
if return_type == "veering_colours":
return veering_colours
elif return_type == "tet_types" or return_type == "num_toggles":
tet_types = []
for tet_num in range(tri.countTetrahedra()):
tet_types.append(tet_type(tri, tet_num, veering_colours))
if return_type == "tet_types":
return tet_types
else:
return tet_types.count("toggle")
elif return_type == "veering_partition":
Llist = []
Rlist = []
for i, direction in enumerate(veering_colours):
if direction == "blue":
Llist.append(i)
else:
Rlist.append(i)
return (Llist, Rlist)
else:
assert return_type == 'boolean'
return True
