Pensieve header: Testing that two half knots match up to a full knot.

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

MatrixSignature[A_]:=Total[Sign[Select[Eigenvalues[A],Abs[#]>(10^(-12))&]]];

(SuperPlus[rs_List]):=Union@@((SuperPlus[#])&/@rs)

(Subscript[Σ, b_])[σ1_,PQ[𝒞1_,q1_]]+(Subscript[Σ, b_])[σ2_,PQ[𝒞2_,q2_]]^:=CF@(Subscript[Σ, b])[σ1+σ2,PQ[𝒞1⋃𝒞2,q1+q2]];
Flip[(Subscript[Σ, B[b___]])[σ_,PQ[𝒞_,q_]]]:=CF@Module[{b1,b2},
b1=Join@@({b}/.i_Integer:>(Subscript[γ, i]));
b2=Join@@(RotateRight/@{b}/.i_Integer:>(Subscript[γ, -i]));
(Subscript[Σ, Reverse/@(B@@(-{b}))])[σ,PQ[𝒞,q]/.(SuperPlus[Thread[b1->b2]])]
]

A[(Subscript[Σ, b_B])[σ_,PQ[𝒞_,q_]]]:=Module[{γs,r,c},
γs=(Subscript[γ, #])&/@Join@@b;
Table[(∂_(r,c))q,{r,(SuperStar[γs])},{c,γs}]
]

pd=PD[Knot[8,5]]

pd1=Take[pd,4]

pd2=Complement[pd,pd1]

TL[pd1]

Flip@TL[pd2]

TL[pd1]+Flip@TL[pd2]

mat=TL[pd1]+Flip@TL[pd2]//A

Plot[MatrixSignature[mat/.ω->(E^(2π I t))],{t,0,1}]

TLSig[pd]

Plot[MatrixSignature[mat/.ω->(E^(2π I t))],{t,0,1}]