This notebook verifies our results on the classification of branching sets of the exceptional-symmetry fillings of the asymmetric L-space SnapPy census knots.
We start with a list containg the symmetric fillings together with their claimed branching sets.
branching_sets_of_symmetric_fillings=[['t12533',[
[(0,1), ['K12n407']],
[(1,1), ['L13n7360(0,0)(2,1)']],
[(-1,1), ['L12n789']],
[(-2,1), ['L12n722(3,1)(0,0)']],
[(-1,2), ['L11n192(-3,1)(0,0)']],
[(-3,1), ['L11n350(0,0)(-4,1)(0,0)']]]],
['t12681',[
[(0,1), ['K11n89']],
[(1,1), ['L11n419(0,0)(5,1)(0,0)', 'L12n1907(0,0)(3,1)(0,0)', 'L13n7356(0,0)(0,0)(2,1)']],
[(-1,1), ['L11n172']],
[(-1,2), ['L10a81(-3,1)(0,0)']],
[(-1,3), ['L11a282(-4, 1)(0,0)']],
[(-2,3), ['L13n4363(-5,3)(0,0)']]]],
['o9_38928',[
[(0,1), ['L11n178']],
[(1,1), ['K11n172']],
[(-1,1), ['L12n703(-3,1)(0,0)']],
[(-2,1), ['L12n1949(0,0)(2, 1)(0,0)']],
[(2,1), ['L13n7433(0,0)(-3, 1)(0,0)', 'L10a105(0,0)(-2,1)', 'L10n54(-8,1)(0,0)']]]],
['o9_39162',[
[(0,1), ['K12n278']],
[(1,1), ['L12n1050']],
[(-1,1), ['L13n9864(-3,1)(0,0)(0,0)']],
[(2,1), ['L10n44(7,1)(0,0)']],
[(1,2), ['L12n784(3,1)(0,0)']],
[(1,4), ['L13n4343(-7,4)(0,0)']]]],
['o9_40363',[
[(0,1), ['K12n479']],
[(1,1), ['L14n43377(0,0)(2, 1)', 'L13n9451(0,0)(3,1)(0,0)', 'L11n419(0,0)(7,1)(0,0)']],
[(-1,1), ['L12n785']],
[(-1,2), ['L11a225(-3,1)(0,0)']],
[(-1,3), ['L12a1861(0,0)(4,1)(0,0)']],
[(-3,4), ['L13n4363(-7,4)(0,0)']]]],
['o9_40487',[
[(0,1), ['L11n152']],
[(1,1), ['K11n147']],
[(-1,1), ['L12n722(-3,1)(0,0)']],
[(1,2), ['L12n968(-3,1)(0,0)']],
[(2,1), ['L11n333(0,0)(2, 1)(0,0)']],
[(3,1), ['L12n1314(7,1)(0,0)']],
[(-2,1), ['L12n1041(-2,1)(0,0)']]]],
['o9_40504',[
[(0,1), ['L11n179']],
[(1,1), ['K11n166']],
[(-1,1), ['L12n715(-3,1)(0,0)']],
[(-2,1), ['L12n1037(-2, 1)(0,0)']],
[(2,1), ['L13n7421(0,0)(-3, 1)(0,0)', 'L10a146(0,0)(-2,1)(0,0)', 'L13n8521(0,0)(5, 1)(0,0)']]]],
['o9_40582',[
[(0,1), ['L13n4413']],
[(1,1), ['K13n2958']],
[(-1,1), ['L10n44(-5,3)(0,0)']],
[(1,2), ['L12n996(3,1)(0,0)']],
[(2,1), ['L13n9833(3,1)(0,0)(0,0)']],
[(4,1), ['L11n350(0,0)(-5,1)(0,0)']]]],
['o9_42675',[
[(0,1), ['L12n702']],
[(1,1), ['L14n24428(5, 3)(0,0)']],
[(-1,1), ['K12n730']],
[(2,1), ['L13n7552(0,0)(-2,1)(0,0)']],
[(-2,1), ['L11n223(-2,1)(0,0)', 'L10n24(0,0)(6,1)', 'L13n9832(0,0)(8,3)(0,0)']]]]]
import snappy
def double_branched_cover(link):
"""
Returns the double branched covers of the link. This works also for links in a more general manifold.
Note that a knot in a more general manifold may have more than one double branched cover
(or no double branched cover at all if the knot represents a primitive element in homology).
This function will return the complete list of all double branched covers of the link.
"""
L=link.copy()
for i in range(L.num_cusps()):
if L.cusp_info(i).filling==(0.0, 0.0):
L.dehn_fill((2,0),i)
return [cov for cov in L.covers(2) if (2.0, 0.0) not in cov.cusp_info('filling')]
def better_is_isometric_to(X,Y,index):
"""
Returns True if X and Y are isometric.
Returns False if X and Y have different homologies. TO DO: Use volume to rigorously distinguish X and Y.
Returns 'unclear' if SnapPy cannot verify it.
The higher the index the harder SnapPy tries.
"""
w='unclear'
if X.homology()!=Y.homology():
w=False
if w=='unclear':
for i in (0,index):
try:
w=X.is_isometric_to(Y)
except RuntimeError:
pass
except snappy.SnapPeaFatalError:
pass
if w==True:
break
if w==False:
w='unclear'
X.randomize()
Y.randomize()
i=i+1
return w
### The following two functions are written by Dunfield and search for positive triangulations.
def all_positive(manifold):
return manifold.solution_type() == 'all tetrahedra positively oriented'
def find_positive_triangulation(manifold, tries=100):
M = manifold.copy()
for i in range(tries):
if all_positive(M):
return M
M.randomize()
for d in M.dual_curves():
X = M.drill(d)
X = X.filled_triangulation()
X.dehn_fill((1,0))
for i in range(tries):
if all_positive(X):
return X
X.randomize()
# In the closed case, here is another trick.
if all(not c for c in M.cusp_info('is_complete')):
for i in range(tries):
# Drills out a random edge
X = M.__class__(M.filled_triangulation())
if all_positive(X):
return X
M.randomize()
def better_find_positive_triangulation(M,tries=1):
'''
Search for a positive triangulation, but ignores errors.
'''
RandomizeCount=0
while RandomizeCount<tries:
try:
X=find_positive_triangulation(M)
return X
except snappy.SnapPeaFatalError:
M.randomize()
RandomizeCount=RandomizeCount+1
return None
def fill_triangulation(M):
'''
Fills all cusps but one.
'''
if M.num_cusps()==1:
return M
M=M.filled_triangulation([0])
M=fill_triangulation(M)
return M
#### This is Dunfield's util.py from his exceptional census
#### for a snappy manifold M descibed as a single filling of a cusp (so do filled_triangulation() as needed)
#### the command regina_name(M) gives what regina identifies M as
"""
This file provides functions for working with Regina (with a little
help from SnapPy) to:
1. Give a standard name ("identify") manifolds, especially Seifert and
graph manifolds.
2. Find essential tori.
3. Try to compute the JSJ decomposition.
"""
import regina
import snappy
import re
import networkx as nx
def appears_hyperbolic(M):
acceptable = ['all tetrahedra positively oriented',
'contains negatively oriented tetrahedra']
return M.solution_type() in acceptable and M.volume() > 0
def children(packet):
child = packet.firstChild()
while child:
yield child
child = child.nextSibling()
def to_regina(data):
if hasattr(data, '_to_string'):
data = data._to_string()
if isinstance(data, str):
if data.find('(') > -1:
data = closed_isosigs(data)[0]
return regina.Triangulation3(data)
assert isinstance(data, regina.Triangulation3)
return data
def extract_vector(surface):
"""
Extract the raw vector of the (almost) normal surface in Regina's
NS_STANDARD coordinate system.
"""
S = surface
T = S.triangulation()
n = T.countTetrahedra()
ans = []
for i in range(n):
for j in range(4):
ans.append(S.triangles(i, j))
for j in range(3):
ans.append(S.quads(i, j))
A = regina.NormalSurface(T, regina.NS_STANDARD, ans)
assert A.sameSurface(S)
return ans
def haken_sum(S1, S2):
T = S1.triangulation()
assert S1.locallyCompatible(S2)
v1, v2 = extract_vector(S1), extract_vector(S2)
sum_vec = [x1 + x2 for x1, x2 in zip(v1, v2)]
A = regina.NormalSurface(T, regina.NS_STANDARD, sum_vec)
assert S1.locallyCompatible(A) and S2.locallyCompatible(A)
assert S1.eulerChar() + S2.eulerChar() == A.eulerChar()
return A
def census_lookup(regina_tri):
"""
Should the input triangulation be in Regina's census, return the
name of the manifold, dropping the triangulation number.
"""
hits = regina.Census.lookup(regina_tri)
hit = hits.first()
if hit is not None:
name = hit.name()
match = re.search('(.*) : #\d+$', name)
if match:
return match.group(1)
else:
return match
def standard_lookup(regina_tri):
match = regina.StandardTriangulation.isStandardTriangulation(regina_tri)
if match:
return match.manifold()
def closed_isosigs(snappy_manifold, trys=20, max_tets=50):
"""
Generate a slew of 1-vertex triangulations of a closed manifold
using SnapPy.
>>> M = snappy.Manifold('m004(1,2)')
>>> len(closed_isosigs(M, trys=5)) > 0
True
"""
M = snappy.Manifold(snappy_manifold)
assert M.cusp_info('complete?') == [False]
surgery_descriptions = [M.copy()]
try:
for curve in M.dual_curves():
N = M.drill(curve)
N.dehn_fill((1,0), 1)
surgery_descriptions.append(N.filled_triangulation([0]))
except snappy.SnapPeaFatalError:
pass
if len(surgery_descriptions) == 1:
# Try again, but unfill the cusp first to try to find more
# dual curves.
try:
filling = M.cusp_info(0).filling
N = M.copy()
N.dehn_fill((0, 0), 0)
N.randomize()
for curve in N.dual_curves():
D = N.drill(curve)
D.dehn_fill([filling, (1,0)])
surgery_descriptions.append(D.filled_triangulation([0]))
except snappy.SnapPeaFatalError:
pass
ans = set()
for N in surgery_descriptions:
for i in range(trys):
T = N.filled_triangulation()
if T._num_fake_cusps() == 1:
n = T.num_tetrahedra()
if n <= max_tets:
ans.add((n, T.triangulation_isosig(decorated=False)))
N.randomize()
return [iso for n, iso in sorted(ans)]
def best_match(matches):
"""
Prioritize the most concise description that Regina provides to
try to avoid things like the Seifert fibered space of a node being
a solid torus or having several nodes that can be condensed into a
single Seifert fibered piece.
"""
def score(m):
if isinstance(m, regina.SFSpace):
s = 0
elif isinstance(m, regina.GraphLoop):
s = 1
elif isinstance(m, regina.GraphPair):
s = 2
elif isinstance(m, regina.GraphTriple):
s = 3
elif m is None:
s = 10000
else:
s = 4
return (s, str(m))
return min(matches, key=score)
def identify_with_torus_boundary(regina_tri):
"""
Use the combined power of Regina and SnapPy to try to give a name
to the input manifold.
"""
kind, name = None, None
P = regina_tri.clone()
P.finiteToIdeal()
P.intelligentSimplify()
M = snappy.Manifold(P.isoSig())
M.simplify()
if appears_hyperbolic(M):
for i in range(100):
if M.solution_type() == 'all tetrahedra positively oriented':
break
M.randomize()
if not M.verify_hyperbolicity(bits_prec=100):
raise RuntimeError('Cannot prove hyperbolicity for ' +
M.triangulation_isosig())
kind = 'hyperbolic'
ids = M.identify()
if ids:
name = ids[0].name()
else:
match = standard_lookup(regina_tri)
if match is None:
Q = P.clone()
Q.idealToFinite()
Q.intelligentSimplify()
match = standard_lookup(Q)
if match is not None:
kind = match.__class__.__name__
name = str(match)
else:
name = P.isoSig()
return kind, name
def is_toroidal(regina_tri):
"""
Checks for essential tori and returns the pieces of the
associated partial JSJ decomposition.
>>> T = to_regina('hLALAkbccfefgglpkusufk') # m004(4,1)
>>> is_toroidal(T)[0]
True
>>> T = to_regina('hvLAQkcdfegfggjwajpmpw') # m004(0,1)
>>> is_toroidal(T)[0]
True
>>> T = to_regina('nLLLLMLPQkcdgfihjlmmlkmlhshnrvaqtpsfnf') # 5_2(10,1)
>>> T.isHaken()
True
>>> is_toroidal(T)[0]
False
Note: currently checks all fundamental normal tori; possibly
the theory lets one just check *vertex* normal tori.
"""
T = regina_tri
assert T.isZeroEfficient()
surfaces = regina.NNormalSurfaceList.enumerate(T,
regina.NS_QUAD, regina.NS_FUNDAMENTAL)
for i in range(surfaces.size()):
S = surfaces.surface(i)
if S.eulerChar() == 0:
if not S.isOrientable():
S = S.doubleSurface()
assert S.isOrientable()
X = S.cutAlong()
X.intelligentSimplify()
X.splitIntoComponents()
pieces = list(children(X))
if all(not C.hasCompressingDisc() for C in pieces):
ids = [identify_with_torus_boundary(C) for C in pieces]
return (True, sorted(ids))
return (False, None)
def decompose_along_tori(regina_tri):
"""
First, finds all essential normal tori in the manifold associated
with fundamental normal surfaces. Then takes a maximal disjoint
collection of these tori, namely the one with the fewest tori
involved, and cuts the manifold open along it. It tries to
identify the pieces, removing any (torus x I) components.
Returns: (has essential torus, list of pieces)
Note: This may fail to be the true JSJ decomposition because there
could be (torus x I)'s in the list of pieces and it might well be
possible to amalgamate some of the pieces into a single SFS.
"""
T = regina_tri
assert T.isZeroEfficient()
essential_tori = []
surfaces = regina.NNormalSurfaceList.enumerate(T,
regina.NS_QUAD, regina.NS_FUNDAMENTAL)
for i in range(surfaces.size()):
S = surfaces.surface(i)
if S.eulerChar() == 0:
if not S.isOrientable():
S = S.doubleSurface()
assert S.isOrientable()
X = S.cutAlong()
X.intelligentSimplify()
X.splitIntoComponents()
pieces = list(children(X))
if all(not C.hasCompressingDisc() for C in pieces):
essential_tori.append(S)
if len(essential_tori) == 0:
return False, None
D = nx.Graph()
for a, A in enumerate(essential_tori):
for b, B in enumerate(essential_tori):
if a < b:
if A.disjoint(B):
D.add_edge(a, b)
cliques = list(nx.find_cliques(D))
if len(cliques) == 0:
clique = [0]
else:
clique = min(cliques, key=len)
clique = [essential_tori[c] for c in clique]
A = clique[0]
for B in clique[1:]:
A = haken_sum(A, B)
X = A.cutAlong()
X.intelligentSimplify()
X.splitIntoComponents()
ids = [identify_with_torus_boundary(C) for C in list(children(X))]
# Remove products
ids = [i for i in ids if i[1] not in ('SFS [A: (1,1)]', 'A x S1')]
return (True, sorted(ids))
def regina_name(closed_snappy_manifold, trys=100):
"""
>>> regina_name('m004(1,0)')
'S3'
>>> regina_name('s006(-2, 1)')
'SFS [A: (5,1)] / [ 0,-1 | -1,0 ]'
>>> regina_name('m010(-1, 1)')
'L(3,1) # RP3'
>>> regina_name('m022(-1,1)')
'SFS [S2: (3,2) (3,2) (4,-3)]'
>>> regina_name('v0004(0, 1)')
'SFS [S2: (2,1) (4,1) (15,-13)]'
>>> regina_name('m305(1, 0)')
'L(3,1) # RP3'
"""
M = snappy.Manifold(closed_snappy_manifold)
isosigs = closed_isosigs(M, trys=trys, max_tets=25)
if len(isosigs) == 0:
return
T = to_regina(isosigs[0])
if T.isIrreducible():
if T.countTetrahedra() <= 11:
for i in range(3):
T.simplifyExhaustive(i)
name = census_lookup(T)
if name is not None:
return name
matches = [standard_lookup(to_regina(iso)) for iso in isosigs]
match = best_match(matches)
if match is not None:
return str(match)
else:
T.connectedSumDecomposition()
pieces = [regina_name(P) for P in children(T)]
if None not in pieces:
return ' # '.join(sorted(pieces))
if __name__ == '__main__':
import doctest
print(doctest.testmod())
<>:84: DeprecationWarning: invalid escape sequence \d
<>:84: DeprecationWarning: invalid escape sequence \d
<ipython-input-3-4bef5675ef00>:84: DeprecationWarning: invalid escape sequence \d
match = re.search('(.*) : #\d+$', name)
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing absolute_igusa_invariants_kohel from here is deprecated. If you need to use it, please import it directly from sage.schemes.hyperelliptic_curves.invariants
See https://trac.sagemath.org/28064 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing absolute_igusa_invariants_wamelen from here is deprecated. If you need to use it, please import it directly from sage.schemes.hyperelliptic_curves.invariants
See https://trac.sagemath.org/28064 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning: this is being removed from the global namespace
See https://trac.sagemath.org/25785 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing all_max_clique from here is deprecated. If you need to use it, please import it directly from sage.graphs.cliquer
See https://trac.sagemath.org/26200 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing backtrack_all from here is deprecated. If you need to use it, please import it directly from sage.games.sudoku_backtrack
See https://trac.sagemath.org/27066 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing berlekamp_massey from here is deprecated. If you need to use it, please import it directly from sage.matrix.berlekamp_massey
See https://trac.sagemath.org/27066 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning: this is being removed from the global namespace
See https://trac.sagemath.org/25785 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing buzzard_tpslopes from here is deprecated. If you need to use it, please import it directly from sage.modular.buzzard
See https://trac.sagemath.org/27066 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing clebsch_invariants from here is deprecated. If you need to use it, please import it directly from sage.schemes.hyperelliptic_curves.invariants
See https://trac.sagemath.org/28064 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing clique_number from here is deprecated. If you need to use it, please import it directly from sage.graphs.cliquer
See https://trac.sagemath.org/26200 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing convergents from here is deprecated. If you need to use it, please import it directly from sage.rings.continued_fraction
See https://trac.sagemath.org/27066 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:511: DeprecationWarning: sage.interacts.debugger is deprecated because it is meant for the deprecated Sage Notebook
See https://trac.sagemath.org/27531 for details.
return hasattr(f, '__wrapped__')
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing designs_from_XML from here is deprecated. If you need to use it, please import it directly from sage.combinat.designs.ext_rep
See https://trac.sagemath.org/27066 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning:
Importing designs_from_XML_url from here is deprecated. If you need to use it, please import it directly from sage.combinat.designs.ext_rep
See https://trac.sagemath.org/27066 for details.
while _is_wrapper(func):
/usr/lib/python3.8/inspect.py:520: DeprecationWarning: this is being removed from the global namespace
See https://trac.sagemath.org/25785 for details.
while _is_wrapper(func):
--------------------------------------------------------------------------- ModuleNotFoundError Traceback (most recent call last) <ipython-input-3-4bef5675ef00> in <module> 350 if __name__ == '__main__': 351 import doctest --> 352 print(doctest.testmod()) /usr/lib/python3.8/doctest.py in testmod(m, name, globs, verbose, report, optionflags, extraglobs, raise_on_error, exclude_empty) 1953 runner = DocTestRunner(verbose=verbose, optionflags=optionflags) 1954 -> 1955 for test in finder.find(m, name, globs=globs, extraglobs=extraglobs): 1956 runner.run(test) 1957 /usr/lib/python3.8/doctest.py in find(self, obj, name, module, globs, extraglobs) 937 # Recursively explore `obj`, extracting DocTests. 938 tests = [] --> 939 self._find(tests, obj, name, module, source_lines, globs, {}) 940 # Sort the tests by alpha order of names, for consistency in 941 # verbose-mode output. This was a feature of doctest in Pythons /usr/lib/python3.8/doctest.py in _find(self, tests, obj, name, module, source_lines, globs, seen) 996 valname = '%s.%s' % (name, valname) 997 # Recurse to functions & classes. --> 998 if ((inspect.isroutine(inspect.unwrap(val)) 999 or inspect.isclass(val)) and 1000 self._from_module(module, val)): /usr/lib/python3.8/inspect.py in unwrap(func, stop) 518 memo = {id(f): f} 519 recursion_limit = sys.getrecursionlimit() --> 520 while _is_wrapper(func): 521 func = func.__wrapped__ 522 id_func = id(func) /usr/lib/python3.8/inspect.py in _is_wrapper(f) 509 if stop is None: 510 def _is_wrapper(f): --> 511 return hasattr(f, '__wrapped__') 512 else: 513 def _is_wrapper(f): /usr/lib/python3/dist-packages/sage/misc/lazy_import.pyx in sage.misc.lazy_import.LazyImport.__getattr__ (build/cythonized/sage/misc/lazy_import.c:3536)() 319 True 320 """ --> 321 return getattr(self.get_object(), attr) 322 323 # We need to wrap all the slot methods, as they are not forwarded /usr/lib/python3/dist-packages/sage/misc/lazy_import.pyx in sage.misc.lazy_import.LazyImport.get_object (build/cythonized/sage/misc/lazy_import.c:2347)() 186 if likely(self._object is not None): 187 return self._object --> 188 return self._get_object() 189 190 cpdef _get_object(self): /usr/lib/python3/dist-packages/sage/misc/lazy_import.pyx in sage.misc.lazy_import.LazyImport._get_object (build/cythonized/sage/misc/lazy_import.c:2586)() 218 elif self._at_startup and not startup_guard: 219 print('Option ``at_startup=True`` for lazy import {0} not needed anymore'.format(self._name)) --> 220 self._object = getattr(__import__(self._module, {}, {}, [self._name]), self._name) 221 name = self._as_name 222 if self._deprecation is not None: ModuleNotFoundError: No module named 'sagenb'
for x in branching_sets_of_symmetric_fillings:
print(x[0])
K=snappy.Manifold(x[0])
for y in x[1]:
K.dehn_fill(y[0])
if y[1]==[]:
print(y[0],'NO BRANCHING SET')
for branching_set in y[1]:
B=snappy.Manifold(branching_set)
F=B.filled_triangulation()
DBC=double_branched_cover(F)
w=False
for D in DBC:
w=better_is_isometric_to(D,K,index=1000)
if w=='unclear':
X=better_find_positive_triangulation(K)
Y=better_find_positive_triangulation(D)
if X and Y is not None:
w=better_is_isometric_to(X,Y,100)
if w=='unclear':
DF=fill_triangulation(D)
K_regina=regina_name(K)
if K_regina is not None and regina_name(DF)==K_regina:
w=True
if w==True:
print(y[0],B,w)
break
if w!=True:
print(y[0],B,'UNCLEAR')
t12533 (0, 1) K12n407(0,0) True (1, 1) L13n7360(0,0)(2,1)(0,0) True (-1, 1) L12n789(0,0)(0,0) True (-2, 1) L12n722(3,1)(0,0) True (-1, 2) L11n192(-3,1)(0,0) True (-3, 1) L11n350(0,0)(-4,1)(0,0) True t12681 (0, 1) K11n89(0,0) True (1, 1) L11n419(0,0)(5,1)(0,0) True (1, 1) L12n1907(0,0)(3,1)(0,0) True (1, 1) L13n7356(0,0)(0,0)(2,1) True (-1, 1) L11n172(0,0)(0,0) True (-1, 2) L10a81(-3,1)(0,0) True (-1, 3) L11a282(-4,1)(0,0) True (-2, 3) L13n4363(-5,3)(0,0) True o9_38928 (0, 1) L11n178(0,0)(0,0) True (1, 1) K11n172(0,0) True (-1, 1) L12n703(-3,1)(0,0) True (-2, 1) L12n1949(0,0)(2,1)(0,0) True (2, 1) L13n7433(0,0)(-3,1)(0,0) True (2, 1) L10a105(0,0)(-2,1) True (2, 1) L10n54(-8,1)(0,0) True o9_39162 (0, 1) K12n278(0,0) True (1, 1) L12n1050(0,0)(0,0) True (-1, 1) L13n9864(-3,1)(0,0)(0,0) True (2, 1) L10n44(7,1)(0,0) True (1, 2) L12n784(3,1)(0,0) True (1, 4) L13n4343(-7,4)(0,0) True o9_40363 (0, 1) K12n479(0,0) True (1, 1) L14n43377(0,0)(2,1)(0,0) True (1, 1) L13n9451(0,0)(3,1)(0,0) True (1, 1) L11n419(0,0)(7,1)(0,0) True (-1, 1) L12n785(0,0)(0,0) True (-1, 2) L11a225(-3,1)(0,0) True (-1, 3) L12a1861(0,0)(4,1)(0,0) True (-3, 4) L13n4363(-7,4)(0,0) True o9_40487 (0, 1) L11n152(0,0)(0,0) True (1, 1) K11n147(0,0) True (-1, 1) L12n722(-3,1)(0,0) True (1, 2) L12n968(-3,1)(0,0) True (2, 1) L11n333(0,0)(2,1)(0,0) True (3, 1) L12n1314(7,1)(0,0) True (-2, 1) L12n1041(-2,1)(0,0) True o9_40504 (0, 1) L11n179(0,0)(0,0) True (1, 1) K11n166(0,0) True (-1, 1) L12n715(-3,1)(0,0) True (-2, 1) L12n1037(-2,1)(0,0) True (2, 1) L13n7421(0,0)(-3,1)(0,0) True (2, 1) L10a146(0,0)(-2,1)(0,0) True (2, 1) L13n8521(0,0)(5,1)(0,0) True o9_40582 (0, 1) L13n4413(0,0)(0,0) True (1, 1) K13n2958(0,0) True (-1, 1) L10n44(-5,3)(0,0) True (1, 2) L12n996(3,1)(0,0) True (2, 1) L13n9833(3,1)(0,0)(0,0) True (4, 1) L11n350(0,0)(-5,1)(0,0) True o9_42675 (0, 1) L12n702(0,0)(0,0) True (1, 1) L14n24428(5,3)(0,0) True (-1, 1) K12n730(0,0) True (2, 1) L13n7552(0,0)(-2,1)(0,0) True (-2, 1) L11n223(-2,1)(0,0) True (-2, 1) L10n24(0,0)(6,1) True (-2, 1) L13n9832(0,0)(8,3)(0,0) True
This confirms that the double branched cover along the claimed branching sets are homeomorphic to the corresponding symmetric fillings.