In [3]:
L.link().sage_link().plot()
Out[3]:
In this notebook we will consider various Hopf RGB links and create the knots K_B^{0,n} and K_G^{0,n} for small values of n. Then we will compute various knot invariants of these knots and in particular distinguish them. We will also verify that the knots really share four surgeries.
First we load the necessary functions.
import snappy
import time
### Find positive triangulations
def all_positive(manifold):
'''
Checks if the solution type of a triangulation is positive.
'''
return manifold.solution_type() == 'all tetrahedra positively oriented'
def find_positive_triangulations(manifold,number=1,tries=100):
'''
Searches for one triangulation with a positive solution type.
(Or if number is set to a different value also for different such triangulations.)
'''
M = manifold.copy()
pos_triangulations=[]
for i in range(tries):
if all_positive(M):
pos_triangulations.append(M)
if len(pos_triangulations)==number:
return pos_triangulations
break
M.randomize()
for d in M.dual_curves(max_segments=500):
X = M.drill(d)
X = X.filled_triangulation()
X.dehn_fill((1,0),-1)
for i in range(tries):
if all_positive(X):
pos_triangulations.append(X)
if len(pos_triangulations)==number:
return pos_triangulations
break
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):
pos_triangulations.append(X)
if len(pos_triangulations)==number:
return pos_triangulations
break
M.randomize()
return pos_triangulations
### CHECKING FOR ISOMETRIES
def better_is_isometric_to(X,Y,return_isometries=False,index=100,try_hard=False):
"""
Returns True if X and Y are isometric.
Returns False if SnapPy could not verify it.
"""
if return_isometries==False:
if X.homology()!=Y.homology():
return False
for i in (0,index):
try:
w=X.is_isometric_to(Y)
if w:
return w
except (RuntimeError,snappy.SnapPeaFatalError):
pass
X.randomize()
Y.randomize()
if try_hard:
pos_triang_X=find_positive_triangulations(X,number=2,tries=index)
pos_triang_Y=find_positive_triangulations(Y,number=2,tries=index)
for X in pos_triang_X:
for Y in pos_triang_Y:
w=better_is_isometric_to(X,Y,index=100,try_hard=False)
if w==True:
return w
return False
if return_isometries==True:
if X.homology()!=Y.homology():
return []
for i in (0,index):
try:
w=X.is_isometric_to(Y,return_isometries)
if w!=False:
return w
except (RuntimeError,snappy.SnapPeaFatalError):
pass
X.randomize()
Y.randomize()
if try_hard:
pos_triang_X=find_positive_triangulations(X,number=2,tries=index)
pos_triang_Y=find_positive_triangulations(Y,number=2,tries=index)
for X in pos_triang_X:
for Y in pos_triang_Y:
w=better_is_isometric_to(X,Y,return_isometries,index=100,try_hard=False)
if w!=[]:
return w
return []
def orientation_preserving_isometric(M,N,index=100,try_hard=False):
'''
Searches for an orientation preserving isometry.
'''
for c in M.dual_curves(max_segments=500):
P=M.drill(c)
for d in N.dual_curves(max_segments=500):
Q=N.drill(d)
for isom in better_is_isometric_to(P,Q,return_isometries=True,index=index,try_hard=try_hard):
if isom.extends_to_link():
if isom.cusp_maps()[0].determinant()==1:
return True
return False
### Invariants
def better_volume(M,index=100,try_hard=False):
'''Computes the verified volume. Returns 0 if SnapPy could not do it.'''
count=0
while count<index:
try:
return M.volume(verified=True)
except:
M.randomize()
count=count+1
if try_hard==True:
pos_triang=find_positive_triangulations(M,number=1,tries=index)
for X in pos_triang:
vol=better_volume(X,index)
if vol!=0:
return vol
return 0
def better_find_canonical_triangulation(M,index=100,try_hard=False):
'''Computes the canonical triangulation. Returns None if SnapPy could not do it.'''
count=0
while count<index:
try:
return M.canonical_retriangulation()
except:
M.randomize()
count=count+1
if try_hard==True:
pos_triang=find_positive_triangulations(M,number=1,tries=index)
for X in pos_triang:
can=better_find_canonical_triangulation(X,index)
if can!=None:
return can
return None
def better_create_diagram(M,index=100,try_hard=False):
'''Creates a knot diagram. Returns None if SnapPy could not do it.'''
count=0
while count<index:
try:
return M.exterior_to_link()
except:
M.randomize()
count=count+1
if try_hard==True:
pos_triang=find_positive_triangulations(M,number=1,tries=index)
for X in pos_triang:
D=better_create_diagram(X,index)
if D!=None:
return D
return None
def subgroups_of_order_n(M,n):
'''
Returns the number of subgroups of order n of the fundamental group of M.
Warning: Works only fast for small n.
'''
return len(snappy.Manifold(M).covers(n))
def subgroups_up_to_order_k(M,k):
'''
Returns the vector containing the number of subgroups of order n of the fundamental group of M for n=2,...,k.
Warning: Works only fast for small k.
'''
vector=[]
for n in range(2,k+1):
vector.append(subgroups_of_order_n(M,n))
return vector
def better_symmetry_group(M,index=100,num_pos_triang=1):
'''
This function computes the symmetry group of the input manifold.
If the second entry is True it is proven to be the symmetry group.
'''
full=False
S='unclear'
try:
S=M.symmetry_group()
full=S.is_full_group()
if full:
return (S,full)
except (ValueError, RuntimeError, snappy.SnapPeaFatalError):
pass
pos_triang=find_positive_triangulations(M,tries=index,number=num_pos_triang)
if pos_triang==[]:
randomizeCount=0
while randomizeCount<index and full==False:
try:
S=M.symmetry_group()
full=S.is_full_group()
if full:
return (S,full)
M.randomize()
randomizeCount=randomizeCount+1
except (ValueError, RuntimeError, snappy.SnapPeaFatalError):
M.randomize()
randomizeCount=randomizeCount+1
for X in pos_triang:
randomizeCount=0
while randomizeCount<index and full==False:
try:
S=X.symmetry_group()
full=S.is_full_group()
if full:
return (S,full)
X.randomize()
randomizeCount=randomizeCount+1
except (ValueError, RuntimeError, snappy.SnapPeaFatalError):
X.randomize()
randomizeCount=randomizeCount+1
return (S,full)
def better_cusp_areas(A,index=100,try_hard=False):
'''Computes the verified cusp areas. Returns 0 if SnapPy could not do it.'''
M=A.high_precision()
count=0
while count<index:
try:
return M.cusp_areas(verified=True)
except:
M.randomize()
count=count+1
if try_hard==True:
pos_triang=find_positive_triangulations(M,number=1,tries=index)
for X in pos_triang:
vol=better_cusp_areas(X,index)
if vol!=0:
return vol
return 0
#### Hopf RGB links
def create_pairs(RBG,k,m):
'''
Takes as input a Hopf RGB link with components 0=red, 1=blue and 2=green and two integers k<m.
It creates the pairs of knots (K_B^{0,n}, K_G^{0,n}) for k<n<m and returns them as a list.
'''
pairs=[]
for n in range(k+1,m):
L.dehn_fill([(0,1),(n,1),(0,0)])
KG=L.filled_triangulation()
KG.set_peripheral_curves( [[[KG.homological_longitude()[1], 0], [KG.homological_longitude()[0], KG.homological_longitude()[1]]]])
KG.set_name('KG_'+str(n))
L.dehn_fill([(0,1),(0,0),(n,1)])
KB=L.filled_triangulation()
KB.set_peripheral_curves( [[[KB.homological_longitude()[1], 0], [KB.homological_longitude()[0], KB.homological_longitude()[1]]]])
KB.set_name('KB_'+str(n))
pairs.append([KB,KG])
return pairs
We start with the example Hopf RBG link from Figure 6.
L=snappy.Manifold('MainExampleHopfRBGLink.lnk')
L.link().sage_link().plot()
First we create the knots K_B^{0,n} and K_G^{0,n} for n in [-10,10]:
pairs=create_pairs(L,-21,21)
pairs
[[KB_-20(0,0), KG_-20(0,0)], [KB_-19(0,0), KG_-19(0,0)], [KB_-18(0,0), KG_-18(0,0)], [KB_-17(0,0), KG_-17(0,0)], [KB_-16(0,0), KG_-16(0,0)], [KB_-15(0,0), KG_-15(0,0)], [KB_-14(0,0), KG_-14(0,0)], [KB_-13(0,0), KG_-13(0,0)], [KB_-12(0,0), KG_-12(0,0)], [KB_-11(0,0), KG_-11(0,0)], [KB_-10(0,0), KG_-10(0,0)], [KB_-9(0,0), KG_-9(0,0)], [KB_-8(0,0), KG_-8(0,0)], [KB_-7(0,0), KG_-7(0,0)], [KB_-6(0,0), KG_-6(0,0)], [KB_-5(0,0), KG_-5(0,0)], [KB_-4(0,0), KG_-4(0,0)], [KB_-3(0,0), KG_-3(0,0)], [KB_-2(0,0), KG_-2(0,0)], [KB_-1(0,0), KG_-1(0,0)], [KB_0(0,0), KG_0(0,0)], [KB_1(0,0), KG_1(0,0)], [KB_2(0,0), KG_2(0,0)], [KB_3(0,0), KG_3(0,0)], [KB_4(0,0), KG_4(0,0)], [KB_5(0,0), KG_5(0,0)], [KB_6(0,0), KG_6(0,0)], [KB_7(0,0), KG_7(0,0)], [KB_8(0,0), KG_8(0,0)], [KB_9(0,0), KG_9(0,0)], [KB_10(0,0), KG_10(0,0)], [KB_11(0,0), KG_11(0,0)], [KB_12(0,0), KG_12(0,0)], [KB_13(0,0), KG_13(0,0)], [KB_14(0,0), KG_14(0,0)], [KB_15(0,0), KG_15(0,0)], [KB_16(0,0), KG_16(0,0)], [KB_17(0,0), KG_17(0,0)], [KB_18(0,0), KG_18(0,0)], [KB_19(0,0), KG_19(0,0)], [KB_20(0,0), KG_20(0,0)]]
Next we compare the various knot invariants and in particular show that the canonical triangulations and the homflypt polynomials (and sometimes the number of covers) can distinguish these knots, while all other standard invariants seem to agree.
### Comparing the canonical triangulations.
start_time=time.time()
different=[]
same=[]
unclear=[]
for [B,G] in pairs:
CB=better_find_canonical_triangulation(B,try_hard=True)
CG=better_find_canonical_triangulation(G,try_hard=True)
if CB!=None:
if CG!=None:
if CB.num_tetrahedra()==CG.num_tetrahedra():
if CB.isomorphisms_to(CG)==[]:
different.append([B,G])
else:
same.append([B,G])
else:
different.append([B,G])
else:
unclear.append([B,G])
else:
unclear.append([B,G])
print('Knots where the canonical triangulations are the same:',same)
print('Knots where the canonical triangulations could not be computed:',unclear)
print('Time taken:',(time.time()-start_time)/60,' minutes')
Knots where the canonical triangulations are the same: [[KB_-1(0,0), KG_-1(0,0)], [KB_0(0,0), KG_0(0,0)], [KB_1(0,0), KG_1(0,0)]] Knots where the canonical triangulations could not be computed: [] Time taken: 0.0007259805997212727 minutes
### Comparing the Alexander polynomials.
start_time=time.time()
different=[]
same=[]
unclear=[]
for [B,G] in pairs:
aB=B.alexander_polynomial()
aG=G.alexander_polynomial()
if aB==aG:
same.append([B,G])
else:
different.append([B,G])
print('Number of knots where the Alexander polynomials are equal:',len(same))
print('Knots where the Alexander polynomials are different:',len(different))
print('Time taken:',(time.time()-start_time)/60,' minutes')
Number of knots where the Alexander polynomials are equal: 41 Knots where the Alexander polynomials are different: 0 Time taken: 0.00478131373723348 minutes
### Comparing the symmetry groups.
start_time=time.time()
different=[]
same=[]
unclear=[]
for [B,G] in pairs:
SB=better_symmetry_group(B)
SG=better_symmetry_group(G)
if SB[1]==True:
if SG[1]==True:
if SB[0].order()==SG[0].order():
same.append([B,G])
else:
different.append([B,G])
else:
unclear.append([B,G])
else:
unclear.append([B,G])
print('Number of knots where the symmetry groups are the same:',len(same))
print('Knots with different symmetry groups:',different)
print('Knots where the symmetry group could not be computed:',unclear)
print('Time taken:',(time.time()-start_time)/60,' minutes')
Number of knots where the symmetry groups are the same: 41 Knots with different symmetry groups: [] Knots where the symmetry group could not be computed: [] Time taken: 0.0005512317021687825 minutes
### Comparing the volumes.
start_time=time.time()
different=[]
same=[]
unclear=[]
for [B,G] in pairs:
volB=better_volume(B,try_hard=True)
volG=better_volume(G,try_hard=True)
if volB!=0:
if volG!=0:
if volB.overlaps(volG):
same.append([B,G])
else:
different.append([B,G])
else:
unclear.append([B,G])
else:
unclear.append([B,G])
print('Number of knots where the volumes overlap:',len(same))
print('Knots with different volumes:',different)
print('Knots where the volumes could not be computed:',unclear)
print('Time taken:',(time.time()-start_time)/60,' minutes')
Number of knots where the volumes overlap: 41 Knots with different volumes: [] Knots where the volumes could not be computed: [] Time taken: 0.04754587014516195 minutes
### Comparing the cusp areas.
start_time=time.time()
different=[]
same=[]
unclear=[]
for [B,G] in pairs:
volB=better_cusp_areas(B,try_hard=True)
volG=better_cusp_areas(G,try_hard=True)
if volB!=0:
if volG!=0:
if volB[0].overlaps(volG[0]):
same.append([B,G])
else:
different.append([B,G])
else:
unclear.append([B,G])
else:
unclear.append([B,G])
print('Number of knots where the cusp areas overlap:',len(same))
print('Knots with different cusp areas:',different)
print('Knots where the cusp areas could not be computed:',unclear)
print('Time taken:',(time.time()-start_time)/60,' minutes')
Number of knots where the cusp areas overlap: 41 Knots with different cusp areas: [] Knots where the cusp areas could not be computed: [] Time taken: 0.222506316502889 minutes
### Comparing the number of covers of degree d.
start_time=time.time()
d=3
print('We look at covers of degree:',d)
different=[]
same=[]
for [B,G] in pairs:
sB=subgroups_of_order_n(B,d)
sG=subgroups_of_order_n(G,d)
if sB==sG:
same.append([B,G])
else:
different.append([B,G])
print('Number of knots where the number of covers agree:',len(same))
print('Knots with different numbers of covers:',different)
print('Time taken:',(time.time()-start_time)/60,' minutes')
We look at covers of degree: 3 Number of knots where the number of covers agree: 41 Knots with different numbers of covers: [] Time taken: 0.003764617443084717 minutes
### Comparing the number of covers of degree d.
start_time=time.time()
d=4
print('We look at covers of degree:',d)
different=[]
same=[]
for [B,G] in pairs:
sB=subgroups_of_order_n(B,d)
sG=subgroups_of_order_n(G,d)
if sB==sG:
same.append([B,G])
else:
different.append([B,G])
print('Number of knots where the number of covers agree:',len(same))
print('Knots with different numbers of covers:',different)
print('Time taken:',(time.time()-start_time)/60,' minutes')
We look at covers of degree: 4 Number of knots where the number of covers agree: 41 Knots with different numbers of covers: [] Time taken: 0.004775861899058024 minutes
### Comparing the number of covers of degree d.
start_time=time.time()
d=5
print('We look at covers of degree:',d)
different=[]
same=[]
for [B,G] in pairs:
sB=subgroups_of_order_n(B,d)
sG=subgroups_of_order_n(G,d)
if sB==sG:
same.append([B,G])
else:
different.append([B,G])
print('Number of knots where the number of covers agree:',len(same))
print('Knots with different numbers of covers:',different)
print('Time taken:',(time.time()-start_time)/60,' minutes')
We look at covers of degree: 5 Number of knots where the number of covers agree: 29 Knots with different numbers of covers: [[KB_-20(0,0), KG_-20(0,0)], [KB_-18(0,0), KG_-18(0,0)], [KB_-16(0,0), KG_-16(0,0)], [KB_-8(0,0), KG_-8(0,0)], [KB_-6(0,0), KG_-6(0,0)], [KB_-4(0,0), KG_-4(0,0)], [KB_4(0,0), KG_4(0,0)], [KB_6(0,0), KG_6(0,0)], [KB_8(0,0), KG_8(0,0)], [KB_16(0,0), KG_16(0,0)], [KB_18(0,0), KG_18(0,0)], [KB_20(0,0), KG_20(0,0)]] Time taken: 0.04880199829737345 minutes
### Create diagrams
start_time=time.time()
pairs=create_pairs(L,-8,8)
diagrams=[]
unclear=[]
for [B,G] in pairs:
DB=better_create_diagram(B,try_hard=True)
DG=better_create_diagram(G,try_hard=True)
DB.simplify('global')
DG.simplify('global')
if DB!=None:
if DG!=None:
diagrams.append([B,DB,G,DG])
else:
unclear.append([B,G])
else:
unclear.append([B,G])
print('Knots where we could not find a diagram:',unclear)
print('Time taken:',(time.time()-start_time)/60,' minutes')
Knots where we could not find a diagram: [] Time taken: 3.0687294801076255 minutes
diagrams
[[KB_-7(0,0), <Link: 1 comp; 240 cross>, KG_-7(0,0), <Link: 1 comp; 161 cross>], [KB_-6(0,0), <Link: 1 comp; 136 cross>, KG_-6(0,0), <Link: 1 comp; 169 cross>], [KB_-5(0,0), <Link: 1 comp; 122 cross>, KG_-5(0,0), <Link: 1 comp; 115 cross>], [KB_-4(0,0), <Link: 1 comp; 102 cross>, KG_-4(0,0), <Link: 1 comp; 95 cross>], [KB_-3(0,0), <Link: 1 comp; 76 cross>, KG_-3(0,0), <Link: 1 comp; 75 cross>], [KB_-2(0,0), <Link: 1 comp; 56 cross>, KG_-2(0,0), <Link: 1 comp; 55 cross>], [KB_-1(0,0), <Link: 1 comp; 34 cross>, KG_-1(0,0), <Link: 1 comp; 35 cross>], [KB_0(0,0), <Link: 1 comp; 12 cross>, KG_0(0,0), <Link: 1 comp; 12 cross>], [KB_1(0,0), <Link: 1 comp; 35 cross>, KG_1(0,0), <Link: 1 comp; 36 cross>], [KB_2(0,0), <Link: 1 comp; 56 cross>, KG_2(0,0), <Link: 1 comp; 57 cross>], [KB_3(0,0), <Link: 1 comp; 76 cross>, KG_3(0,0), <Link: 1 comp; 78 cross>], [KB_4(0,0), <Link: 1 comp; 102 cross>, KG_4(0,0), <Link: 1 comp; 99 cross>], [KB_5(0,0), <Link: 1 comp; 116 cross>, KG_5(0,0), <Link: 1 comp; 117 cross>], [KB_6(0,0), <Link: 1 comp; 168 cross>, KG_6(0,0), <Link: 1 comp; 143 cross>], [KB_7(0,0), <Link: 1 comp; 250 cross>, KG_7(0,0), <Link: 1 comp; 156 cross>]]
### Comparing the HOMFLYPT polynomials.
start_time=time.time()
var('v')
different=[]
same=[]
unclear=[]
for [B,DB,G,DG] in diagrams:
aB=DB.sage_link().homfly_polynomial(normalization='vz')
print(B)
print(aB(v,0).expand())
aG=DG.sage_link().homfly_polynomial(normalization='vz')
print(G)
print(aG(v,0).expand())
if aB==aG:
same.append([B,G])
else:
different.append([B,G])
print('------------')
print('Knots where the HOMFLYPT polynomials are equal:',same)
print('Number of knots where the HOMFLYPT polynomials are different:',len(different))
print('Time taken:',(time.time()-start_time)/60,' minutes')
KB_-7(0,0) 63*v^36 - 68*v^34 - 506*v^32 + 1451*v^30 - 1582*v^28 + 797*v^26 - 155*v^24 - 16*v^20 + 60*v^18 - 85*v^16 + 55*v^14 - 15*v^12 + v^10 + 1 KG_-7(0,0) 84*v^36 - 194*v^34 - 192*v^32 + 1035*v^30 - 1273*v^28 + 675*v^26 - 135*v^24 - 9*v^20 + 26*v^18 - 19*v^16 - 9*v^14 + 16*v^12 - 5*v^10 + 1 ------------ KB_-6(0,0) 48*v^32 - 52*v^30 - 374*v^28 + 1063*v^26 - 1145*v^24 + 569*v^22 - 109*v^20 - 14*v^18 + 52*v^16 - 73*v^14 + 47*v^12 - 13*v^10 + v^8 + 1 KG_-6(0,0) 63*v^32 - 142*v^30 - 150*v^28 + 767*v^26 - 926*v^24 + 483*v^22 - 95*v^20 - 8*v^18 + 23*v^16 - 17*v^14 - 7*v^12 + 13*v^10 - 4*v^8 + 1 ------------ KB_-5(0,0) 35*v^28 - 38*v^26 - 262*v^24 + 735*v^22 - 778*v^20 + 379*v^18 - 83*v^16 + 44*v^14 - 61*v^12 + 39*v^10 - 11*v^8 + v^6 + 1 KG_-5(0,0) 45*v^28 - 98*v^26 - 113*v^24 + 539*v^22 - 634*v^20 + 323*v^18 - 69*v^16 + 20*v^14 - 15*v^12 - 5*v^10 + 10*v^8 - 3*v^6 + 1 ------------ KB_-4(0,0) 24*v^24 - 26*v^22 - 170*v^20 + 467*v^18 - 481*v^16 + 217*v^14 - 5*v^12 - 49*v^10 + 31*v^8 - 9*v^6 + v^4 + 1 KG_-4(0,0) 30*v^24 - 62*v^22 - 81*v^20 + 351*v^18 - 397*v^16 + 189*v^14 - 19*v^12 - 13*v^10 - 3*v^8 + 7*v^6 - 2*v^4 + 1 ------------ KB_-3(0,0) 15*v^20 - 16*v^18 - 98*v^16 + 259*v^14 - 262*v^12 + 141*v^10 - 56*v^8 + 23*v^6 - 7*v^4 + v^2 + 1 KG_-3(0,0) 18*v^20 - 34*v^18 - 54*v^16 + 203*v^14 - 220*v^12 + 113*v^10 - 28*v^8 - v^6 + 4*v^4 - v^2 + 1 ------------ KB_-2(0,0) 8*v^16 - 8*v^14 - 46*v^12 + 105*v^10 - 77*v^8 + 12*v^6 + 10*v^4 - 5*v^2 + 2 KG_-2(0,0) 9*v^16 - 14*v^14 - 32*v^12 + 91*v^10 - 77*v^8 + 26*v^6 - 4*v^4 + v^2 + 1 ------------ KB_-1(0,0) 3*v^12 - 2*v^10 - 18*v^8 + 35*v^6 - 23*v^4 + 6*v^2 + 1/v^2 - 1 KG_-1(0,0) 3*v^12 - 2*v^10 - 18*v^8 + 35*v^6 - 23*v^4 + 6*v^2 + 1/v^2 - 1 ------------ KB_0(0,0) 2*v^4 - 6*v^2 - 2/v^2 + 7 KG_0(0,0) 2*v^4 - 6*v^2 - 2/v^2 + 7 ------------ KB_1(0,0) -v^4 + 18/v^2 - 45/v^4 + 38/v^6 - 11/v^8 + 2 KG_1(0,0) -v^4 + 18/v^2 - 45/v^4 + 38/v^6 - 11/v^8 + 2 ------------ KB_2(0,0) 2*v^2 + 27/v^2 - 55/v^4 + 122/v^6 - 168/v^8 + 113/v^10 - 29/v^12 - 11 KG_2(0,0) -15/v^2 + 15/v^4 + 52/v^6 - 126/v^8 + 99/v^10 - 27/v^12 + 3 ------------ KB_3(0,0) -20/v^2 + 38/v^4 - 23/v^6 - 81/v^8 + 272/v^10 - 362/v^12 + 227/v^14 - 55/v^16 + 5 KG_3(0,0) -4/v^2 + 10/v^4 - 23/v^6 - 11/v^8 + 160/v^10 - 278/v^12 + 195/v^14 - 50/v^16 + 2 ------------ KB_4(0,0) 6/v^2 - 28/v^4 + 47/v^6 - 25/v^8 + 5/v^10 - 153/v^12 + 483/v^14 - 625/v^16 + 379/v^18 - 89/v^20 + 1 KG_4(0,0) 2/v^2 - 7/v^4 + 3/v^6 + 31/v^8 - 79/v^10 + 1/v^12 + 287/v^14 - 481/v^16 + 323/v^18 - 80/v^20 + 1 ------------ KB_5(0,0) 8/v^4 - 36/v^6 + 59/v^8 - 41/v^10 + 24/v^12 - 7/v^14 - 242/v^16 + 755/v^18 - 958/v^20 + 569/v^22 - 131/v^24 + 1 KG_5(0,0) 3/v^4 - 10/v^6 + 5/v^8 + 15/v^10 + 10/v^12 - 91/v^14 - 18/v^16 + 459/v^18 - 739/v^20 + 483/v^22 - 117/v^24 + 1 ------------ KB_6(0,0) 10/v^6 - 44/v^8 + 71/v^10 - 49/v^12 + 11/v^14 + 25/v^16 - 16/v^18 - 350/v^20 + 1087/v^22 - 1361/v^24 + 797/v^26 - 181/v^28 + 1 KG_6(0,0) 4/v^6 - 13/v^8 + 7/v^10 + 17/v^12 - 23/v^14 + 53/v^16 - 142/v^18 - 36/v^20 + 671/v^22 - 1052/v^24 + 675/v^26 - 161/v^28 + 1 ------------ KB_7(0,0) 12/v^8 - 52/v^10 + 83/v^12 - 57/v^14 + 13/v^16 + 1/v^18 + 35/v^20 - 26/v^22 - 478/v^24 + 1479/v^26 - 1834/v^28 + 1063/v^30 - 239/v^32 + 1 KG_7(0,0) 5/v^8 - 16/v^10 + 9/v^12 + 19/v^14 - 26/v^16 + 9/v^18 + 63/v^20 - 194/v^22 - 59/v^24 + 923/v^26 - 1420/v^28 + 899/v^30 - 212/v^32 + 1 ------------ Knots where the HOMFLYPT polynomials are equal: [[KB_-1(0,0), KG_-1(0,0)], [KB_0(0,0), KG_0(0,0)], [KB_1(0,0), KG_1(0,0)]] Number of knots where the HOMFLYPT polynomials are different: 12 Time taken: 0.08436941703160604 minutes
### Comparing the knot floer homologies.
start_time=time.time()
different=[]
same=[]
unclear=[]
for [B,DB,G,DG] in diagrams[4:-4]:
aB=DB.knot_floer_homology()
aG=DG.knot_floer_homology()
if aB==aG:
same.append([B,G])
else:
different.append([B,G])
print('Number of knots where the knot Floer homology invariants are equal:',len(same))
print('Knots where the knot Floer homology invariants are different:',different)
print('Time taken:',(time.time()-start_time)/60,' minutes')
Number of knots where the knot Floer homology invariants are equal: 7 Knots where the knot Floer homology invariants are different: [] Time taken: 0.30374369621276853 minutes
### Comparing the signatures.
start_time=time.time()
different=[]
same=[]
unclear=[]
for [B,DB,G,DG] in diagrams[4:-4]:
aB=DB.signature()
aG=DG.signature()
if aB==aG:
same.append([B,G])
else:
different.append([B,G])
print('Number of knots where the signatures are equal:',len(same))
print('Knots where the signatures are different:',different)
print('Time taken:',(time.time()-start_time)/60,' minutes')
Number of knots where the signatures are equal: 7 Knots where the signatures are different: [] Time taken: 0.0009354472160339356 minutes
Similarly, we can check more complicated Hopf RBG links. It is also interesting that we can do the same for Hopf RBG links with fewer ribbon intersections. A concrete example is the following.
L=snappy.Manifold('3strand_working_complicated.lnk')
L.link().sage_link().plot()
pairs=create_pairs(L,-11,11)
pairs
[[KB_-10(0,0), KG_-10(0,0)], [KB_-9(0,0), KG_-9(0,0)], [KB_-8(0,0), KG_-8(0,0)], [KB_-7(0,0), KG_-7(0,0)], [KB_-6(0,0), KG_-6(0,0)], [KB_-5(0,0), KG_-5(0,0)], [KB_-4(0,0), KG_-4(0,0)], [KB_-3(0,0), KG_-3(0,0)], [KB_-2(0,0), KG_-2(0,0)], [KB_-1(0,0), KG_-1(0,0)], [KB_0(0,0), KG_0(0,0)], [KB_1(0,0), KG_1(0,0)], [KB_2(0,0), KG_2(0,0)], [KB_3(0,0), KG_3(0,0)], [KB_4(0,0), KG_4(0,0)], [KB_5(0,0), KG_5(0,0)], [KB_6(0,0), KG_6(0,0)], [KB_7(0,0), KG_7(0,0)], [KB_8(0,0), KG_8(0,0)], [KB_9(0,0), KG_9(0,0)], [KB_10(0,0), KG_10(0,0)]]
### Comparing the canonical triangulations.
start_time=time.time()
different=[]
same=[]
unclear=[]
for [B,G] in pairs:
CB=better_find_canonical_triangulation(B,try_hard=True)
CG=better_find_canonical_triangulation(G,try_hard=True)
if CB!=None:
if CG!=None:
if CB.num_tetrahedra()==CG.num_tetrahedra():
if CB.isomorphisms_to(CG)==[]:
different.append([B,G])
else:
same.append([B,G])
else:
different.append([B,G])
else:
unclear.append([B,G])
else:
unclear.append([B,G])
print('Knots where the canonical triangulations are the same:',same)
print('Knots where the canonical triangulations could not be computed:',unclear)
print('Time taken:',(time.time()-start_time)/60,' minutes')
Knots where the canonical triangulations are the same: [[KB_-1(0,0), KG_-1(0,0)], [KB_0(0,0), KG_0(0,0)], [KB_1(0,0), KG_1(0,0)]] Knots where the canonical triangulations could not be computed: [] Time taken: 0.00893728733062744 minutes
We also verify that the constructed Legendrian links are really different.
L=snappy.Manifold('Legendrian_Hopf_RBG.lnk')
L.link().sage_link().plot()
L.dehn_fill([(0,1),(-2,1),(0,0)])
B=L.filled_triangulation()
B
unnamed link_filled(0,0)
L.dehn_fill([(0,1),(0,0),(-2,1)])
G=L.filled_triangulation()
G
unnamed link_filled(0,0)
B.is_isometric_to(G)
False
CB=B.canonical_retriangulation()
CG=G.canonical_retriangulation()
CG.num_tetrahedra()
22
CB.num_tetrahedra()
21
DB=B.exterior_to_link()
DB
<Link: 1 comp; 72 cross>
DG=G.exterior_to_link()
DG
<Link: 1 comp; 61 cross>
var('v')
v
DG.sage_link().homfly_polynomial(normalization='vz')(v,0).expand()
9*v^22 - 14*v^20 - 32*v^18 + 91*v^16 - 77*v^14 + 26*v^12 - 4*v^10 + v^6 + v^2
DB.sage_link().homfly_polynomial(normalization='vz')(v,0).expand()
8*v^22 - 8*v^20 - 46*v^18 + 105*v^16 - 77*v^14 + 12*v^12 + 10*v^10 - 6*v^8 + 2*v^6 + v^2