CodeLines.txt

def Delta(lista,B):
    res=B(1)
    l=copy(lista)[:-1]
    while len(l)>0:
        res=res*B(l)
        l=l[:-1]
    return res

def clean_conj(lista,F):
    n=len(lista)
    if n%2==0:
        print "Not conjugate to a generator"
        return None
    m=ZZ((n-1)/2)
    a=lista[m]
    b=lista[m+1:]
    if F(b)*F(lista[:m])!=F(1):
        print "Not conjugate to a generator"
        return None
    return [a,b]        
  
def list_of_relations_2(orden0,wiring0):
    wiring=[[B(_[0]),_[1]]for _ in wiring0]
    res1=[]
    trenza=B(1)
    orden=copy(orden0)
    for cruce in wiring:
        u,v=cruce
        orden=(u^-1).permutation()*orden
        trenza=trenza*u
        w=[orden.inverse()(_) for _ in v]
        res1.append([trenza,v,w,orden])
        trw=Delta(w,B)
        trenza=trenza*trw
        orden=trw.permutation().inverse()*orden
    final=[]
    for aa in res1:
        lis=[]
        for i in aa[2]:
            accion=[_.sign()*orden0(_.abs()) for _ in (F([i])*aa[0]^-1).Tietze()]
            listas=clean_conj(accion,F)
            if listas==None:
                print 'Problems'
                return None
            else:
                lis+=[listas]
        datos1=[[_[0] for _ in lis],[_[1] for _ in lis]]
        final+=[datos1]
    return final
    
 
def CommProd1(a,l1,F,LR,XX):
    ll1=list(F(l1).Tietze())
    if len(ll1)==0:
        return M(0)
    return XX[a,ll1[0]]+tt([ll1[0]],LR)*CommProd1(a,ll1[1:],F,LR,XX)


def CommCyclic1(L,F,LR,XX,dir='RL'):
    res=[]
    r=len(L[0])
    comb=L[0]
    mn=comb.index(min(comb))
    L1=[v[mn:]+v[:mn] for v in L]
    LF=[F([-u for u in reversed(L1[1][j])]+[L1[0][j]]+L1[1][j]) for j in range(r)]
    for j in range(1,r):
        u=F(L1[1][j])
        if dir=='RL':
            LFp=[_ for _ in reversed(LF[j+1:]+LF[:j])]
        elif dir=='LR':
            LFp=LF[j+1:]+LF[:j]
        w=u*prod(LFp)*u^-1
        res+=[CommProd1(L1[0][j],list(w.Tietze()),F,LR,XX)]
    return res
    
def tt(i,ring):
    return prod([LR.gen(ZZ(j).abs()-1)^(ZZ(j).sign()) for j in i])
    
def MatTrunc(mat):
    A=mat
    m=A.nrows()
    return Matrix(m,[v.truncate(2) for v in A.list()])
    
    
def EscalonarM2(matriz,n):
    U=matriz
    U0=Matrix(U.nrows(),[v.constant_coefficient() for v in U.list()])
    A,B=U0.echelon_form(transformation=True)
    U1=B*U
    Apivot=[]
    j=0
    for i in range(A.rank()):
        while A[i,j]==0:
            j=j+1
        Apivot.append(j)
    for i in range(A.rank()):
        j=Apivot[i]
        U1.rescale_row(i,U1[i,j]^-1)
        for k in range(i)+range(i+1,A.nrows()):
            U1.add_multiple_of_row(k,i,-U1[k,j])
    SustNeg=identity_matrix(T0,n*(n-1)/2)
    for i in range(A.rank()):
        for  j in range(n*(n-1)/2):
            SustNeg[j,Apivot[i]]=-U1[i,j].truncate(2).polynomial()
    SustNeg=SustNeg.delete_rows(Apivot)
    return SustNeg

def ImagenMorfismo(L,XX,YY,S,LRv,dicv,inc):
    i,j=L
    res=XX[L].change_ring(S)
    ai=tt([i],LRv).subs(dicv)
    aj=tt([j],LRv).subs(dicv)
    aij=(ai*aj).truncate(2)
    bij=(ai-aij)
    cij=(aj-aij)
    res+=sum([(bij*YY[i,u,v]-cij*YY[j,u,v])*XX[u,v].change_ring(S) for (u,v) in inc])
    return res
       	

ordenPos=Permutation([1, 4, 9, 5, 10, 11, 2, 7, 8, 6, 3])
wiringPos=[[(), [7, 8]], [(), [5, 10]], [(), [5, 11, 2, 8]], [(), [5, 7]], [(),
[5, 6]], [(), [5, 3]], [(-7, -8), [9, 10]], [(), [9, 8]], [(), [9, 2]],
[(), [9, 7, 6]], [(), [9, 11, 3]], [(-8, -4, 7), [6, 11]], [(-7, -5, -6,
-4, 8), [4, 10, 11]], [(), [4, 2]], [(), [4, 7]], [(), [4, 8, 3, 6]],
[(3,), [10, 7]], [(4, 5, 2, 3, 4, -2), [1, 10, 2, 6]], [(), [1, 11, 7]],
[(), [1, 3]], [(), [1, 8]], [(-3, -4), [10, 3]]]
ordenNeg=Permutation([1, 9, 5, 4, 2, 7, 11, 10, 3, 6, 8])
wiringNeg=[[(), [4, 2]], [(), [4, 7]], [(), [4, 11, 10]], [(), [4, 3, 6, 8]],
[(-5, -6, -7, -4, -5), [5, 10]], [(), [5, 11, 2, 8]], [(), [5, 7]], [(),
[5, 6]], [(), [5, 3]], [(-4, 3, 6), [11, 6]], [(-6, -5), [8, 7]], [(4,
5, 4, 8, -7, 6), [3, 10]], [(7,), [9, 2]], [(), [9, 7, 6]], [(), [9,
10]], [(), [9, 8]], [(), [9, 3, 11]], [(-4, 5, -6, -3), [1, 2, 10, 6]],
[(), [1, 8]], [(), [1, 11, 7]], [(), [1, 3]], [(-2, -4), [10, 7]]]
n=11
B=BraidGroup(n)
F=FreeGroup(n)
       	

GrupoListaPos=list_of_relations_2(ordenPos,wiringPos)
GrupoListaNeg=list_of_relations_2(ordenNeg,wiringNeg)
combPos=[sorted(v[0]) for v in GrupoListaPos]
combNeg=[sorted(v[0]) for v in GrupoListaNeg]
print "Do combinatorics coincide? ",sorted(combPos)==sorted(combNeg)
# Construct the list inc as a basis of commutators in G_1/G_2
comb=[]
for v in combPos:
    for j in v[1:]:
        comb.append([v[0],j])
tot=[tuple(sorted(_.list())) for _ in Subsets([1..n],2)] 
inc=[_ for _ in tot if list(_) not in comb]

# The variables of the ring S are the unknowns of the final integral equation
xx=var(['v%da%db%d'% (k,i,j) for k in [1..n] for (i,j) in inc])
S=PolynomialRing(ZZ,xx)

# The dictionnary YY relates the index of the unknowns with the morphism of groups.
YY={}
for k in [1..n]:
    for l in range(len(inc)):
        YY[k,inc[l][0],inc[l][1]]=S.gen((k-1)*len(inc)+l)

LR=LaurentPolynomialRing(ZZ,'t',n)
LRv=LR.change_ring(S)
R=LR.polynomial_ring()
T=PowerSeriesRing(S,'s',num_gens=n,default_prec=2)
T0=T.change_ring(ZZ)
HomLRT0=R.hom([1+v for v in T0.gens()],codomain=T0,check=True)
dic={v:HomLRT0(v) for v in LR.gens()}
dicv={v:HomLRT0(v).change_ring(S) for v in LRv.gens()}

#We work with several modules, with basis the set of commutators. The elements XX[i,j] gives the commutator (any sign and order) with respect to the ring of Laurent polynomials

M=FreeModule(LR,n*(n-1)/2)
#MT=M.change_ring(T0)
#Mv=M.change_ring(T)
#XX={(i,j):M.gen(n*(i-1)-i*(i+1)/2+j-1) for i in range(1,n) for j in range(i+1,n+1)}
XX={tot[i]:M.gen(i) for i in range(n*(n-1)/2)}
for i in range(1,n+1):
    XX[i,i]=M(0)
    XX[-i,-i]=M(0)
    XX[i,-i]=M(0)
    XX[-i,i]=M(0)
    for j in range(i+1,n+1):
        XX[-i,j]=-tt([i],LR)^-1*XX[i,j]
        XX[i,-j]=-tt([j],LR)^-1*XX[i,j]
        XX[-i,-j]=tt([i],LR)^-1*tt([j],LR)^-1*XX[i,j]
        XX[j,i]=-XX[i,j]
        XX[-j,i]=-XX[i,-j]
        XX[j,-i]=-XX[-i,j]
        XX[-j,-i]=-XX[-i,-j]

# Translate the relations in the groups as relations for the Alexander invariants
relAlexPos=[]
for v in GrupoListaPos:
    relAlexPos+=CommCyclic1(v,F,LR,XX,dir='LR')
print "Alexander invariant for first group"
relAlexNeg=[]
for v in GrupoListaNeg:
    relAlexNeg+=CommCyclic1(v,F,LR,XX,dir='LR')
nrels=len(relAlexNeg)
print "Alexander invariant for the second group"

# The above relations are written in the power series ring truncated at level 2
relAlexSeriePos=Matrix([vector(w.subs(dic) for w in v.list()) for v in relAlexPos])
relAlexSerieNeg=Matrix([vector(w.subs(dic) for w in v.list()) for v in relAlexNeg])

#This matrix codifies each commutator in terms of the basis inc
SustNeg=EscalonarM2(relAlexSerieNeg,n)
print "The relations are used to write every one in terms of the basis inc"

# Image of each commutator of the first group in the second group
TotImagenMorfismo=[]
for u in tot:
    v=ImagenMorfismo(u,XX,YY,S,LRv,dicv,inc)
    TotImagenMorfismo.append(v)
    if u[-1]==n:
        print "Finished pairs with line ", u[0]
        
relAlexVarPos=[]
for ser in relAlexSeriePos:
    vct=0
    for j in range(n*(n-1)/2):
        vct+=ser[j].change_ring(S)*TotImagenMorfismo[j]
    relAlexVarPos.append(vct)
relAlexVarPos=MatTrunc(Matrix(relAlexVarPos))
relAlexVarPos=MatTrunc(relAlexVarPos*SustNeg.change_ring(T).transpose())
print "Images of relations of first group in terms of the basis inc in the second group, done"

relZPos=[]
for prueba in relAlexVarPos:
    relZPos.append(vector(flatten([[pr1.derivative(v).constant_coefficient() for v in T.gens()] for pr1 in prueba])))
relZPos=Matrix(nrels,relZPos)
print "Integral matrix from relations"


#Adding Jacobi relations in the base inc and in the associated one in the next level
JCB=[]
for i in range(1,n-1):
    for j in range(i+1,n):
        for k in range(j+1,n+1):
            JCB+=[T0.gen(i-1)*XX[j,k].change_ring(T0)+T0.gen(j-1)*XX[k,i].change_ring(T0)+T0.gen(k-1)*XX[i,j].change_ring(T0)]
JCB=Matrix(JCB)
JCB1=MatTrunc(JCB*SustNeg.transpose())
JCB2=Matrix(JCB.nrows(),flatten([[v.derivative(w).constant_coefficient() for w in T0.gens()] for v in JCB1.list()]))
print "Simplified Jacobi relations as integral matrix done"

#Eliminating generators from Jacobi relations (if no torsion)
SF,U,V=JCB2.smith_form()
Jdiag=[SF[v,v] for v in range(SF.rank())]
if Set(Jdiag)!=Set([1]):
    print "Torsion at level 2"
print "Smith form of Jacobi relations, done", SF.ncols()-SF.rank()," generators left"

# Reduce the relations taking into account Jacobi relations (no torsion!)
eqs=(relZPos*V).matrix_from_columns([SF.rank()..JCB2.ncols()-1])
eqs1=[_ for _ in list(Set(eqs.list())) if _!=0]
print len(eqs1)," equations in ",len(inc)*n," unknowns."

# Put the system in matrix for, print the diagonal of the Smith form and a particular solution
Aeq=Matrix([vector(eq.monomial_coefficient(v) for v in S.gens()) for eq in eqs1])
Beq=vector(-eq.constant_coefficient() for eq in eqs1)
print "Ranks: ", Aeq.rank(),Aeq.augment(Beq).rank()
U1,U2,U3=Aeq.smith_form()
diageq=[U1[v,v] for v in range(U1.rank())]
diageq1=[_ for _ in diageq if _!=1]
print "Ones in the diagonal of the smith form: ",diageq.count(1)
print "Rest of the diagonal of the smith form: ",diageq1
print "Non integers in the Particular Solution:", [_ for _ in U1.solve_right(U2*Beq) if _ not in ZZ]
       	

  

ordenPos1=copy(ordenPos)
wiringPos1=[[tuple(-i for i in _[0]),_[1]] for _ in wiringPos]
       	
  

GrupoListaPos1=list_of_relations_2(ordenPos1,wiringPos1)
combPos1=[sorted(v[0]) for v in GrupoListaPos1]
print "Do combinatorics coincide? ",sorted(combPos1)==sorted(combNeg)
# Construct the list inc as a basis of commutators in G_1/G_2

# Translate the relations in the groups as relations for the Alexander invariants
relAlexPos1=[]
for v in GrupoListaPos1:
    relAlexPos1+=CommCyclic1(v,F,LR,XX,dir='LR')
print "Alexander invariant for first group"

# The above relations are written in the power series ring truncated at level 2
relAlexSeriePos1=Matrix([vector(w.subs(dic) for w in v.list()) for v in relAlexPos1])


        
relAlexVarPos1=[]
for ser in relAlexSeriePos1:
    vct=0
    for j in range(n*(n-1)/2):
        vct+=ser[j].change_ring(S)*TotImagenMorfismo[j]
    relAlexVarPos1.append(vct)
relAlexVarPos1=MatTrunc(Matrix(relAlexVarPos1))
relAlexVarPos1=MatTrunc(relAlexVarPos1*SustNeg.change_ring(T).transpose())
print "Images of relations of first group in terms of the basis inc in the second group, done"

relZPos1=[]
for prueba in relAlexVarPos1:
    relZPos1.append(vector(flatten([[pr1.derivative(v).constant_coefficient() for v in T.gens()] for pr1 in prueba])))
relZPos1=Matrix(nrels,relZPos1)
print "Integral matrix from relations"


# Reduce the relations taking into account Jacobi relations (no torsion!)
eqs_1=(relZPos1*V).matrix_from_columns([SF.rank()..JCB2.ncols()-1])
eqs1_1=[_ for _ in list(Set(eqs_1.list())) if _!=0]
print len(eqs1_1)," equations in ",len(inc)*n," unknowns."

# Put the system in matrix for, print the diagonal of the Smith form and a particular solution
Aeq1=Matrix([vector(eq.monomial_coefficient(v) for v in S.gens()) for eq in eqs1_1])
Beq1=vector(-eq.constant_coefficient() for eq in eqs1_1)
print "Ranks: ", Aeq1.rank(),Aeq1.augment(Beq1).rank()
U1_1,U2_1,U3_1=Aeq1.smith_form()
diageq_1=[U1_1[v,v] for v in range(U1_1.rank())]
diageq1_1=[_ for _ in diageq_1 if _!=1]
print "Ones in the diagonal of the smith form: ",diageq_1.count(1)
print "Rest of the diagonal of the smith form: ",diageq1_1
print "Non integers in the Particular Solution:", [_ for _ in U1_1.solve_right(U2_1*Beq1) if _ not in ZZ]