In this notebook we will take the census L-space knots wogether with their stable coefficients and determine which integer slopes in the alternating interval admit OSBs and if so return the corresponding Goeritz matrices.
##This is code that will search for an obtuse superbase
##in an integer CM lattice. Input is a list of
##coefficients for CM lattice.
from itertools import combinations
import numpy as np
import time
import csv
def norm(v):
return sum([z*z for z in v])
def irred_check(v,vtest):
#Computes T=(v-vtest).vtest returns true if this does not prove
#that v is reducible. I.e returns true if T<0 or vtest=0 or v=vtest.
T=sum([(v[i]-vtest[i])*vtest[i] for i in range(0,len(v))])
if T<0:
return True
if T==0:
if sum([z*z for z in vtest])==0:
return True
if sum([(v[i]-vtest[i])*(v[i]-vtest[i]) for i in range(0,len(v))])==0:
return True
return False
def irred_bounds(sigma):
#Provide bounds on the coefficients of an irreducible
#in a changemaker lattice.
n=len(sigma)
vmax=[0]*n
#Check if sigma is tight
tight=False
for i in range(1,n):
if sigma[i]> sum([sigma[j] for j in range(0,i)]):
tight=True
break
if tight:
vmax[0]=2
else:
vmax[0]=1
for i in range(1, len(sigma)):
if sigma[i]==sigma[i-1]:
vmax[i]=vmax[i-1]
else:
#If sigma[i] tight.
if sigma[i]> sum([sigma[j] for j in range(0,i)]):
vmax[i]=6+sum([vmax[j]+1 for j in range(0,i)])
vmax[i]=min(vmax[i], sigma[i]*(sigma[i]+2))
else:
T=sigma[i]
j=i-1
while T>0:
j=max([k for k in range(0,j+1) if sigma[k]<=T])
vmax[i]=vmax[i]+vmax[j]+1
T-=sigma[j]
#print(vmax)
return(vmax)
def irred_generator(sigma, vmax):
n=len(sigma)
v=[-vmax[i] for i in range(0,n)]
k=0
while True:
#Check whether there is a choice final coordinate to
#put vector in CM lattice. If so, yield that vector.
t= sum([sigma[i]*v[i] for i in range(0,n-1)])
if t%sigma[-1]==0:
v[-1]=-t//sigma[-1]
if v[-1]>=-vmax[-1] and v[-1]<=vmax[-1]:
yield v
#Now increment v.
while k<n-1 and v[k]+1>vmax[k]:
v[k]=-vmax[k]
k=k+1
if k<n-1:
v[k]+=1
k=0
else:
break
def list_irreducibles(sigma, quick=False):
##generate a list of irreducible elements of a CM lattice.
#This version uses the generator function irred_generator.
p=norm(sigma)
n=len(sigma)
irred_list=[]
#Vector defining the bounds on the search space
vmax=irred_bounds(sigma)
if quick:
vmax=[min(2, vmax[i]) for i in range(0,n)]
for v in irred_generator(sigma,vmax):
v_irred_flag=True
for vtest in irred_list:
if irred_check(v,vtest)==False:
v_irred_flag=False
break
#if v is potentially irreducible, we use it filter out any
#non-irreducibles in irred_list and add it to irred_list.
if v_irred_flag:
irred_list=[w for w in irred_list if irred_check(w,v)]
irred_list+=[list(v)]
return irred_list
def OSB_search(sigma, quick=False, verbose=False):
##Search for an obtuse superbase. Function returns an OSB
#if it finds one. Returns false otherwise. The optional flag quick
##can be used to accelerate the search. When true the flag restricts
##the search of irreducibles to vectors with coefficients at most 2.
##Conjecturally, this is should still always find an OSB if it exists.
##
##This implementation is improved
## adding norm 2 vertices to the basis at the start
##
if verbose:
print("------")
if quick:
print("Accelerated search.")
else:
print("Standard search.")
print("sigma="+ str(sigma))
start_time = time.time()
p=norm(sigma)
n=len(sigma)
#Create a list of norm 2 vectors that we will put in our basis.
norm2_vertices=[]
for i in range(1,n):
if sigma[i]==sigma[i-1]:
z=[0]*n
z[i]=1
z[i-1]=-1
norm2_vertices+=[list(z)]
#Filter our set of irreducibles by picking those with norm >2 and
#pairing 0 or -1 with the norm two vectors
vertex_pool=[]
for z in list_irreducibles(sigma,quick):
if len(norm2_vertices)==0 and norm(z)>0:
vertex_pool+=[z]
elif norm(z)>2:
t=[sum([w[i]*z[i] for i in range(0,n)]) for w in norm2_vertices]
if min(t)>=-1 and max(t)<=0:
vertex_pool+=[z]
#Now consider all possible combinations of vertices.
#Uses combinations from itertools package.
for t in combinations(vertex_pool,n-len(norm2_vertices)-1):
vertex_list=norm2_vertices+list(t)
M=np.asarray(vertex_list)
#Calculate intersection matrix
Q=np.matmul(M,M.transpose())
#Check that the obtuse conditions are satisfied.
obtuse_flag=True
if min(sum(Q))<0:
obtuse_flag =False
for i in range(0,n-1):
for j in range(0,i):
if Q[i,j]>0:
obtuse_flag=False
break
#Check that this is a basis by computing determinant of Q.
#The det function in numpy doesn't seem to do integer
#calculations.
det=np.linalg.det(Q)
if obtuse_flag and det>p-0.5 and det<p+0.5:
if verbose:
print(vertex_list)
print(Q)
print(det)
print("--- Time taken: %s seconds ---" % (time.time() - start_time))
return vertex_list
if verbose:
print("No OSB found.")
print("--- Time taken: %s seconds ---" % (time.time() - start_time))
return False
def admits_alt_surg(stable_coefs):
##Input the stable coefficients. Return True if
##a CM lattice with these coefficients has an OSB
##returns False otherwise.
start_time = time.time()
stable=sorted(stable_coefs)
print(stable)
for sigma in [[1]*i+[1]*(stable[0]-1)+stable for i in range(0,3)]:
#Check that each sigma satisfies CM condition
n=norm(sigma)
CM_flag=True
for i in range(1,len(sigma)):
if sigma[i]> 1+sum([sigma[j] for j in range(0,i)]):
CM_flag=False
n=norm(sigma)
if CM_flag:
if OSB_search(sigma)!=False:
print(str(n)+"-CM lattice has OSB")
else:
print(str(n)+"-CM lattice has no OSB")
print("--- Time taken: %s seconds ---" % (time.time() - start_time))
#admits_alt_surg([2,2,2,3,6])
#-----
#Benchmark set 1
#sigma=[1,1,3,3,4,5]
#sigma=[1,1,1,3,3,4,5]
#sigma=[1,1,1,1,3,3,4,5]
#-----
#Benchmark set 2
#sigma=[1,2,2,2,3,6]
#sigma=[1,1,2,2,2,3,6]
#sigma=[1,1,1,2,2,2,3,6]
#-----
#sigma=[1,1,1,2,2,6]
#OSB_search(sigma, verbose=True)
#OSB_search(sigma,quick=True, verbose=True)
##This is code that will search for an obtuse superbase
##in an integer CM lattice. Input is a list of
##coefficients for CM lattice.
from itertools import combinations
import numpy as np
import time
import math
def is_tight(sigma):
#Check whether a CM lattice has a tight index.
for i in range(1,len(sigma)):
if sigma[i]==1+ sum([sigma[j] for j in range(0,i)]):
return True
return False
def is_CM(sigma):
##Check the vector satisfies the CM condition
for i in range(1,len(sigma)):
if sigma[i]> 1+sum([sigma[j] for j in range(0,i)]):
return False
return True
def stable_is_decomposable(stable_coefs):
#Check whether any changemaker lattice associated to a set of stable
#coefficients can possibly be decomposable
stable=sorted(stable_coefs)
return is_decomposable([1]*stable[0]+stable)
def is_decomposable(sigma):
#Check whether a given changemaker lattice
#is decomposable.
##Uses Greene's algorithm of calculating the standard
##basis elements and checking that their intersection
##graph is connected.
if is_tight(sigma) or not is_CM(sigma):
return False
STD_basis=[]
n=len(sigma)
for i in range(1,n):
v=[0]*n
v[i]=-1
T=sigma[i]
j=i
while T>0:
j=max([k for k in range(0,j) if sigma[k]<=T])
v[j]=1
T-=sigma[j]
STD_basis.append(list(v))
S1=[STD_basis[0]]
S2=[]
for i in range(1,len(STD_basis)):
v=STD_basis[i]
flag=True
for w in S1:
if sum([v[j]*w[j] for j in range(0,n)])!=0:
flag=False
S1.append(list(v))
break
if flag:
S2.append(list(v))
if len(S2)==0:
return False
for v in S2:
for w in S1:
if sum([v[j]*w[j] for j in range(0,n)])!=0:
return False
return True
def norm(v):
return sum([z*z for z in v])
def irred_check(v,vtest):
#Computes T=(v-vtest).vtest returns true if this does not prove
#that v is reducible. I.e returns true if T<0 or vtest=0 or v=vtest.
T=sum([(v[i]-vtest[i])*vtest[i] for i in range(0,len(v))])
if T<0:
return True
if T==0:
if sum([z*z for z in vtest])==0:
return True
if sum([(v[i]-vtest[i])*(v[i]-vtest[i]) for i in range(0,len(v))])==0:
return True
return False
def vertex_bounds(sigma):
#Provide bounds on the coefficients of a vertex in
#in an OSB in a changemaker lattice.
n=len(sigma)
vmax=[0]*n
p=norm(sigma)
#Check if sigma is tight
vmax[0]=1
if is_tight(sigma):
vmax[0]=2
else:
vmax[0]=1
for i in range(1, len(sigma)):
if sigma[i]==sigma[i-1]:
vmax[i]=vmax[i-1]
continue
if sigma[i-1]==1 and sigma[i]>1:
vmax[i]=sigma[i]
continue
#If sigma[i] tight.
if sigma[i]> sum([sigma[j] for j in range(0,i)]):
vmax[i]=5+sum([vmax[j]+1 for j in range(0,i)])
else:
T=sigma[i]
j=i-1
while T>0:
j=max([k for k in range(0,j+1) if sigma[k]<=T])
vmax[i]=vmax[i]+vmax[j]+1
T-=sigma[j]
#Compute the bound coming from Cauchy-Schwartz inequality
for i in range(1,len(sigma)):
CS_bound= int(math.sqrt(p - sigma[i]*sigma[i]))
vmax[i]=min(CS_bound, vmax[i])
return vmax
def CM_generator(sigma, vmax):
##Iterates through all vectors in a changemaker lattice
##whose coordinates satisfy |v[i]|\leq vmax[i] for all i.
n=len(sigma)
v=[-vmax[i] for i in range(0,n)]
k=0
while True:
#Check whether there is a choice final coordinate to
#put vector in CM lattice. If so, yield that vector.
t= sum([sigma[i]*v[i] for i in range(0,n-1)])
if t%sigma[-1]==0:
v[-1]=-t//sigma[-1]
if v[-1]>=-vmax[-1] and v[-1]<=vmax[-1]:
yield v
#Now increment v.
while k<n-1 and v[k]+1>vmax[k]:
v[k]=-vmax[k]
k=k+1
if k<n-1:
v[k]+=1
k=0
else:
break
def list_vertices(sigma, quick=False):
##generate a list of candidate vertices with norm at least 3.
##If the quick flag is set to true, this does not
##generate precisely the set of all irreducibles.
p=norm(sigma)
n=len(sigma)
irred_list=[]
equal_list=[]
#Vector defining the bounds on the search space
vmax=vertex_bounds(sigma)
#Modify vmax if we are doing a quick search
if quick:
vmax=[min(2, vmax[i]) for i in range(0,n)]
for i in range(0,n):
if sigma[i]==sigma[-1]:
vmax[i]=1
for k in range(1,sigma[-1]+1):
if sigma.count(k)>1:
equal_list.append([i for i in range(0,n) if sigma[i]==k])
for v in CM_generator(sigma,vmax):
#Skip vectors of norm less than 2.
if norm(v)<=2:# or norm_calc>p:
continue
v_irred_flag=True
#Check whether there is a vertex of norm 2 which
#shows reducibility
for clist in equal_list:
mini=min([v[i] for i in clist])
maxi=max([v[i] for i in clist])
if maxi>mini+1:
v_irred_flag=False
break
##Finally compare v against all other potential irreducibles.
if v_irred_flag:
for vtest in irred_list:
if irred_check(v,vtest)==False:
v_irred_flag=False
break
#if v is potentially irreducible, we use it filter out any
#non-irreducibles in irred_list and add it to irred_list.
if v_irred_flag:
irred_list=[w for w in irred_list if irred_check(w,v)]
irred_list+=[list(v)]
return irred_list
def obtuse_check(Q,n):
if min(sum(Q))<0:
return False
for i in range(0,n-1):
for j in range(0,i):
if Q[i,j]>0:
return False
return True
def OSB_search(sigma, quick=False):
##Search for an obtuse superbase. Function returns an OSB
#if it finds one. Returns false otherwise. The optional flag quick
##can be used to accelerate the search. When true the flag restricts
##the search of irreducibles to vectors with coefficients at most 2.
##Conjecturally, this is should still always find an OSB if it exists.
##However, we can only rigourously rule out the existence of
##an OSB if we run the algorithm with the quick flag set to False.
##This implementation is improved
## adding norm 2 vertices to the basis at the start
p=norm(sigma)
n=len(sigma)
#Create a list of norm 2 vectors that we will put in our basis.
norm2_vertices=[]
for i in range(1,n):
if sigma[i]==sigma[i-1]:
z=[0]*n
z[i]=1
z[i-1]=-1
norm2_vertices+=[list(z)]
#Filter our set of irreducibles by picking those with norm >2 and
#pairing 0 or -1 with the norm two vectors
vertex_pool=[]
for z in list_vertices(sigma,quick):
t=[sum([w[i]*z[i] for i in range(0,n)]) for w in norm2_vertices]
if len(t)==0 or (min(t)>=-1 and max(t)<=0):
vertex_pool+=[z]
#Now consider all possible combinations of vertices.
#Uses combinations from itertools package.
for t in combinations(vertex_pool,n-len(norm2_vertices)-1):
vertex_list=norm2_vertices+list(t)
M=np.asarray(vertex_list)
#Calculate intersection matrix
Q=np.matmul(M,M.transpose())
#Check that this is a basis by computing determinant of Q.
#The det function in numpy doesn't seem to do integer
#calculations.
det=np.linalg.det(Q)
if obtuse_check(Q,n) and det>p-0.5 and det<p+0.5:
return Q
return []
def adhoc_tests(stable_coefs):
##Some tests for whether a given set of cable coefficients
##can possibly support an OSB. Returns false if there cannot
##exist an OSB. Returns true if inconclusive. These methods
##are not necessary they are simply used to accelerate running
##times.
stable=sorted(stable_coefs)
##If the stable coefficents contain 4,4,4,4,3,3,2 then
##there is no OSB.
if stable.count(4)>=4 and stable.count(3)>=2 and stable.count(2)>=1:
return False
##If the stable coefficents contain 3,3,3,3,2,2,2 then
##there is no OSB.
if stable.count(3)>=4 and stable.count(2)>=3:
return False
##The obstruction based on Corollary ?? from the paper.
for k in range(0,len(stable)-1):
N=stable.count(stable[k])
if N>=4 and (N-1)*stable[k]>stable[k+1]>stable[k]:
return False
return True
def find_OSB_slopes(stable_coefs):
##Input the stable coefficients. Returns a rigourously verified
##list of coefficients for which there is an OSB along
##with a gram matrix for each OSB.
output=[]
if adhoc_tests(stable_coefs)==False:
return output
stable=sorted(stable_coefs)
##Check whether these stable coefficients support a decomposable lattice.
decomposable= stable_is_decomposable(stable)
slopes=[]
##Work out which slopes can conceivably admit an OSB.
if decomposable:
slopes=[0,1,2]
elif is_CM([1]*(stable[0]-1)+stable):
slopes=[0,1]
else:
slopes=[1,2]
#First perform a quick search to find OSBs with small coefficients.
for k in slopes:
sigma=[1]*k+[1]*(stable[0]-1)+stable
n=norm(sigma)
if is_CM(sigma):
##First run a quick search
result=OSB_search(sigma,quick=True)
if len(result)>0:
output+=[[n,result]]
else:
##If an OSB not found quickly, run a full search instead.
result=OSB_search(sigma, quick=False)
if len(result)>0:
output+=[[n,result]]
return output
def crossing(goeritz_matrix):
'''
Computes the crossing number of an alternating link diagram corresponding to the Goeritz matrix.
'''
return sum([goeritz_matrix[i][i] + sum(goeritz_matrix[i]) for i in range(len(goeritz_matrix))])/2
We load the list of L-space knots in the SnapPy census knots together with their stable coefficients.
L_space_knots_with_stable_coefficients=[]
with open('stable_coefficients.csv', 'r') as file:
reader = csv.reader(file)
for row in reader:
L_space_knots_with_stable_coefficients.append(row)
L_space_knots_with_stable_coefficients=L_space_knots_with_stable_coefficients[1:]
len(L_space_knots_with_stable_coefficients)
508
We compute the Goeritz matrices.
start_time = time.time()
GOERITZ=[]
L_space_knots_without_alternating_surgery=[]
for [knot,stable,N] in L_space_knots_with_stable_coefficients:
goeritz=find_OSB_slopes(eval(stable))
if goeritz==[]:
L_space_knots_without_alternating_surgery.append(knot)
else:
GOERITZ.append([knot, stable,goeritz])
print('Number of knots that have no integer surgery to a manifold with an OSB (and thus also have no alternating surgery):',len(L_space_knots_without_alternating_surgery))
print('Number of knots that have a single integer surgery to a manifold with an OSB:',len([x for x in GOERITZ if len(x[2])==1]))
print('Number of knots that have exactly two integer surgery to a manifold with an OSB:',len([x for x in GOERITZ if len(x[2])==2]))
print('Number of knots that have exactly three integer surgery to a manifold with an OSB:',len([x for x in GOERITZ if len(x[2])==3]))
print("--- Time taken: %s seconds ---" % ((time.time() - start_time)))
Number of knots that have no integer surgery to a manifold with an OSB (and thus also have no alternating surgery): 115 Number of knots that have a single integer surgery to a manifold with an OSB: 12 Number of knots that have exactly two integer surgery to a manifold with an OSB: 381 Number of knots that have exactly three integer surgery to a manifold with an OSB: 0 --- Time taken: 94.0670690536499 seconds ---
Next, we prove the following claim: If an integer alternating surgery along a SnapPy census knots yields a manifold which is the double branched cover over an alternating knot K in the 3-sphere, then K has crossing number at most 13. For that we compute for each Goeritz matrix the crossing number of the corresponding alternating link diagram as explained in the paper and then we return the maximal crossing number.
start_time = time.time()
max_crossing_number=0
for knot in GOERITZ:
for matrix in knot[2]:
if crossing(matrix[1])>max_crossing_number:
max_crossing_number=crossing(matrix[1])
print('The maximal crossing number is:', int(max_crossing_number))
print("--- Time taken: %s seconds ---" % ((time.time() - start_time)))
The maximal crossing number is: 13 --- Time taken: 0.0039098262786865234 seconds ---
We print and save the data:
with open("Goeritz.csv", "a") as output: #Opens output file # Use the a parameter to add a row
csvwriter = csv.writer(output, delimiter = ",")
csvwriter.writerow(('knot','stable coefficients','OSBs'))
for x in GOERITZ:
with open("Goeritz.csv", "a") as output: #Opens output file # Use the a parameter to add a row
csvwriter = csv.writer(output, delimiter = ",")
csvwriter.writerow((x[0],x[1],x[2]))
GOERITZ
[['m016', '[3, 2, 2]', [[18, array([[ 2, 0, -1], [ 0, 4, -3], [-1, -3, 5]])], [19, array([[ 2, 0, -1, 0], [ 0, 2, 0, -1], [-1, 0, 4, -2], [ 0, -1, -2, 3]])]]], ['m071', '[5, 2]', [[31, array([[ 2, -1, 0], [-1, 6, -1], [ 0, -1, 3]])], [32, array([[ 2, -1, 0, -1], [-1, 2, -1, 0], [ 0, -1, 6, -1], [-1, 0, -1, 3]])]]], ['m082', '[3, 3, 2, 2]', [[27, array([[ 2, 0, -1, 0], [ 0, 2, 0, -1], [-1, 0, 5, -3], [ 0, -1, -3, 4]])], [28, array([[ 2, 0, 0, 0, -1], [ 0, 2, 0, -1, 0], [ 0, 0, 2, 0, -1], [ 0, -1, 0, 3, -2], [-1, 0, -1, -2, 4]])]]], ['m103', '[5, 3, 2, 2]', [[43, array([[ 2, -1, 0, -1], [-1, 3, 0, 0], [ 0, 0, 4, -1], [-1, 0, -1, 3]])], [44, array([[ 2, 0, 0, -1, -1], [ 0, 2, -1, 0, -1], [ 0, -1, 3, 0, 0], [-1, 0, 0, 4, -1], [-1, -1, 0, -1, 3]])]]], ['m118', '[4, 3, 2]', [[30, array([[ 4, -2, -1], [-2, 6, -2], [-1, -2, 3]])], [31, array([[ 2, -1, 0, 0], [-1, 4, -2, -1], [ 0, -2, 3, -1], [ 0, -1, -1, 4]])]]], ['m144', '[3, 3, 3, 2, 2]', [[36, array([[ 2, 0, 0, -1, 0], [ 0, 2, -1, 0, 0], [ 0, -1, 2, 0, -1], [-1, 0, 0, 5, -3], [ 0, 0, -1, -3, 4]])], [37, array([[ 2, 0, 0, 0, 0, -1], [ 0, 2, 0, 0, -1, 0], [ 0, 0, 2, -1, 0, 0], [ 0, 0, -1, 2, 0, -1], [ 0, -1, 0, 0, 3, -2], [-1, 0, 0, -1, -2, 4]])]]], ['m194', '[5, 3]', [[36, array([[ 2, -1, -1], [-1, 6, -3], [-1, -3, 6]])], [37, array([[ 2, -1, -1, 0], [-1, 2, 0, 0], [-1, 0, 6, -3], [ 0, 0, -3, 4]])]]], ['m198', '[5, 2, 2, 2]', [[38, array([[ 2, -1, 0, -1], [-1, 2, 0, 0], [ 0, 0, 5, -3], [-1, 0, -3, 5]])], [39, array([[ 2, 0, 0, -1, 0], [ 0, 2, -1, 0, -1], [ 0, -1, 2, 0, 0], [-1, 0, 0, 5, -2], [ 0, -1, 0, -2, 3]])]]], ['m239', '[4, 3, 2, 2]', [[34, array([[ 2, -1, 0, -1], [-1, 4, -1, -1], [ 0, -1, 4, -1], [-1, -1, -1, 3]])], [35, array([[ 2, 0, -1, 0, 0], [ 0, 2, 0, -1, 0], [-1, 0, 3, -1, -1], [ 0, -1, -1, 3, -1], [ 0, 0, -1, -1, 3]])]]], ['m240', '[4, 3, 3]', [[36, array([[ 2, 0, 0, -1], [ 0, 2, 0, -1], [ 0, 0, 5, -4], [-1, -1, -4, 6]])], [37, array([[ 2, -1, 0, 0, 0], [-1, 2, 0, -1, 0], [ 0, 0, 2, 0, -1], [ 0, -1, 0, 5, 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