Pensieve header: An analysis of K15a55264 and K15n90489. Continues PossibleCounterexample.nb at pensieve://Projects/Signatures.

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

K=Mirror@Knot[3,1]
Factor[Alexander[K][T]]
lhs=TLSig[K]
lhs==KasSig[K]

K=Knot[15,Alternating,55264]
Factor[Alexander[K][T]]
lhs=TLSig[K]
lhs==KasSig[K]

sign[ℰ_]:=Echo@Module[{n,d,v,p,rs,e,k},Echo@ℰ;
{n,d}=NumeratorDenominator[ℰ];
{n,d}/=(ω^(Exponent[n,ω]/2+Exponent[n,ω,Min]/2));
p=Factor[ω2[v]@n*ω2[v]@d/.v->4(u^2)-2];
rs=Solve[p==0,u,Reals];
If[rs==={}, Sign[p/.u->0],
rs=Union@(u/.rs);
Sign[(((-1))^(e=Exponent[p,u]))Coefficient[p,u,e]]+Sum[
k=0;While[(d=RootReduce[(∂_({u,++k}))p/.u->r])==0];
If[EvenQ[k],0,2Sign[d]]*θ[u-r],
{r,rs}] ]]

K=Knot[15,NonAlternating,90489]
Factor[Alexander[K][T]]
Print["Computing KasSig..."]
rhs=KasSig[K]
Print["Computing TLSig..."]
TLSig[K]==rhs

sign[ℰ_]:=Module[{n,d,v,p,rs,e,k},
{n,d}=NumeratorDenominator[ℰ];{n,d}/=(ω^(Exponent[n,ω]/2));
p=Factor[ω2[v]@n*ω2[v]@d/.v->4(u^2)-2];
rs=Solve[p==0,u,Reals];
If[rs==={}, Return[Sign[p/.u->0]]];
rs=Union@(u/.rs);
Sign[(((-1))^(e=Exponent[p,u]))Coefficient[p,u,e]]+Sum[
k=0;While[(d=RootReduce[(∂_({u,++k}))p/.u->r])==0];
If[EvenQ[k],0,2Sign[d]]*θ[u-r],
{r,rs}] ]

K=Knot[16,Alternating,144399]
Factor[Alexander[K][T]]
lhs=TLSig[K]
lhs==KasSig[K]

K=Knot[16,NonAlternating,225282]
Factor[Alexander[K][T]]
lhs=TLSig[K]
lhs==KasSig[K]

K=Knot[16,NonAlternating,761158]
Factor[Alexander[K][T]]
lhs=TLSig[K]
lhs==KasSig[K]