SetDirectory["C:\\drorbn\\AcademicPensieve\\Talks\\ICERM-2305"];
<< Signatures`

{(SuperStar[((2u-ω+3(ω^(-1)))(Subscript[(OverBar[η]), 1])(Subscript[η, 2]))]), (SuperPlus[((Subscript[η, 1])->ω (Subscript[η, 2]))])}

CF[{(Subscript[η, 1])-(Subscript[η, 2]),(Subscript[η, 1])-(Subscript[η, 3])}]

RulesOf[(Subscript[η, 1])+(Subscript[η, 2])+(Subscript[η, 3])]

(Subscript[Σ, B[{-1,2}]])[0,PQ[{},0]]⋃(Subscript[Σ, B[{-3,4}]])[0,PQ[{},0]]//(Subscript[FM, -1,4])

Kas/@{(SubPlus[X])[1,2,3,4],(SubMinus[X])[1,4,3,2]}

TL/@{X[1,2,3,4],X[1,4,3,2]}

Kas[Knot[3,1]]

KasSig[Knot[3,1]]

TLSig[Knot[3,1]]

Kas[X[1,5,2,4]]⋃Kas[X[2,5,3,6]]

Kas[X[1,5,2,4]]⋃Kas[X[2,5,3,6]]//(Subscript[FM, 5,-2])

Kas[X[1,5,2,4]]⋃Kas[X[2,5,3,6]]//(Subscript[FM, 5,-2])//(Subscript[Cordon, -2])

Kas[X[1,5,2,4]]⋃Kas[X[2,5,3,6]]//(Subscript[FM, -2,5])//(Subscript[Cordon, 5])//(Subscript[Cordon, -2])

lhs=Kas[X[4,2,5,1]]⋃Kas[X[7,3,8,2]]⋃Kas[X[8,6,9,5]]//mc;
rhs=Kas[X[7,5,8,4]]⋃Kas[X[8,2,9,1]]⋃Kas[X[5,3,6,2]]//mc
{lhs[[1]],rhs[[1]]}
Simplify[lhs[[2,2]]==rhs[[2,2]]]

f=KasSig@PD[X[4,2,5,1],X[7,3,8,2],X[8,6,9,5]];
Plot[f,{u,-1.2,1.2}]

lhs=TL[X[4,2,5,1]]⋃TL[X[7,3,8,2]]⋃TL[X[8,6,9,5]]//mc;
rhs=TL[X[7,5,8,4]]⋃TL[X[8,2,9,1]]⋃TL[X[5,3,6,2]]//mc
{lhs[[1]],rhs[[1]]}
lhs[[2,2]]==rhs[[2,2]]

f=TLSig@PD[X[4,2,5,1],X[7,3,8,2],X[8,6,9,5]];
Plot[f,{u,-1.2,1.2}]

Kashaev for Knots

-KnotSignature/@AllKnots[{3,8}]

(*u=0;*)
Kas[Knot[3,1]]
Clear[u]

(Subscript[Σ, B[]])[sign[(1/2) (3-4 (u^2))]+sign[-2 (-1+2 (u^2))]+sign[-((-3+4 (u^2))/(-1+2 (u^2)))],PQ[{},0]]

Table[K->2KasSig[K],{K,AllKnots[{3,7}]}]//Column

f=KasSig[Knot[10,1]]
Plot[f,{u,-1,1}]

f=TLSig[Knot[10,1]]
Plot[f,{u,-1,1}]

(KasSig/@AllKnots[{3,8}])/.u->1/2

f=KasSig[Knot[9,5]]
Plot[f,{u,-1,1}]

f=TLSig[Knot[9,5]]
Plot[f,{u,-1,1}]

f=KasSig[Knot[8,2]]
Plot[f,{u,-1,1}]

f=TLSig[Knot[8,2]]
Plot[f,{u,-1,1}]

f=KasSig[Knot[12,Alternating,422]]
Plot[f,{u,-1,1}]

f=TLSig[Knot[12,Alternating,422]]
Plot[f,{u,-1,1}]

Tristram-Levine for Knots

-KnotSignature/@AllKnots[{3,8}]

TL[Knot[3,1]]

TLSig/@AllKnots[{3,8}] /. u->0

TLSig/@AllKnots[{3,8}] /. u->0.9999

Table[K->Column[{f=TLSig[K],Plot[f,{u,-1,1}]}, Center],
{K,AllKnots[{3,8}]}]//Column

The Naik-Stanford Double Delta Move

=

pd=PD[(Subscript[X, 6,10,28,24]),(Subscript[(OverBar[X]), 28,3,29,19]),(Subscript[X, 26,20,27,19]),(Subscript[(OverBar[X]), 27,23,11,24]),(Subscript[X, 1,12,13,30]),(Subscript[(OverBar[X]), 13,5,14,25]),(Subscript[X, 17,26,18,25]),(Subscript[(OverBar[X]), 18,29,8,30]),(Subscript[X, 4,7,22,15]),(Subscript[(OverBar[X]), 22,2,23,16]),(Subscript[X, 20,17,21,16]),(Subscript[(OverBar[X]), 21,14,9,15])]/.{(Subscript[X, i_,j_,k_,l_]):>(Subscript[X, -i,j,k,-l]),(Subscript[(OverBar[X]), i_,j_,k_,l_]):>(Subscript[(OverBar[X]), -j,k,l,-i])}
lhs=TL[pd]

pd=PD[(Subscript[X, 5,9,25,21]),(Subscript[(OverBar[X]), 25,4,26,22]),(Subscript[X, 29,23,30,22]),(Subscript[(OverBar[X]), 30,20,12,21]),(Subscript[X, 2,11,16,27]),(Subscript[(OverBar[X]), 16,6,17,28]),(Subscript[X, 14,29,15,28]),(Subscript[(OverBar[X]), 15,26,7,27]),(Subscript[X, 3,8,19,18]),(Subscript[(OverBar[X]), 19,1,20,13]),(Subscript[X, 23,14,24,13]),(Subscript[(OverBar[X]), 24,17,10,18])]/.{(Subscript[X, i_,j_,k_,l_]):>(Subscript[X, -i,j,k,-l]),(Subscript[(OverBar[X]), i_,j_,k_,l_]):>(Subscript[(OverBar[X]), -j,k,l,-i])}
rhs=TL[pd]

lhs==rhs

pd=PD[(Subscript[X, 6,10,28,24]),(Subscript[(OverBar[X]), 28,3,29,19]),(Subscript[X, 26,20,27,19]),(Subscript[(OverBar[X]), 27,23,11,24]),(Subscript[X, 1,12,13,30]),(Subscript[(OverBar[X]), 13,5,14,25]),(Subscript[X, 17,26,18,25]),(Subscript[(OverBar[X]), 18,29,8,30]),(Subscript[X, 4,7,22,15]),(Subscript[(OverBar[X]), 22,2,23,16]),(Subscript[X, 20,17,21,16]),(Subscript[(OverBar[X]), 21,14,9,15])]/.{(Subscript[X, i_,j_,k_,l_]):>(Subscript[X, -i,j,k,-l]),(Subscript[(OverBar[X]), i_,j_,k_,l_]):>(Subscript[(OverBar[X]), -j,k,l,-i])}
lhs=Kas[pd]

pd=PD[(Subscript[X, 5,9,25,21]),(Subscript[(OverBar[X]), 25,4,26,22]),(Subscript[X, 29,23,30,22]),(Subscript[(OverBar[X]), 30,20,12,21]),(Subscript[X, 2,11,16,27]),(Subscript[(OverBar[X]), 16,6,17,28]),(Subscript[X, 14,29,15,28]),(Subscript[(OverBar[X]), 15,26,7,27]),(Subscript[X, 3,8,19,18]),(Subscript[(OverBar[X]), 19,1,20,13]),(Subscript[X, 23,14,24,13]),(Subscript[(OverBar[X]), 24,17,10,18])]/.{(Subscript[X, i_,j_,k_,l_]):>(Subscript[X, -i,j,k,-l]),(Subscript[(OverBar[X]), i_,j_,k_,l_]):>(Subscript[(OverBar[X]), -j,k,l,-i])}
rhs=Kas[pd]

lhs==rhs