Pensieve header: Running through the Bedlewo Program.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Talks\\CMS-2112"]

Once[<< KnotTheory`]

MatrixSignature[A_]:=Total[Sign[Select[Eigenvalues[A],Abs[#]>(10^(-12))&]]];
Writhe[K_]:=Sum[If[PositiveQ[x],1,-1], {x,List@@PD@K}];

<< BedAndKas.m;

PD[(Subscript[8, 2])]=PD[X[10,1,11,2], X[2,11,3,12],X[12,3,13,4],X[4,13,5,14],X[14,5,15,6],X[8,16,9,15],X[16,8,1,7],X[6,9,7,10]];

{lhs=Jones[(Subscript[8, 2])][q], rhs=Jones[Knot[8,2]][q],lhs==rhs}

K=(Subscript[8, 2]);

t=1-ω; r=t+t\[Conjugate];

XingsByArmpits=List@@PD[K]/.x:X[i_,j_,k_,l_]:>If[PositiveQ[x],(SubPlus[X])[-i,j,k,-l],(SubMinus[X])[-j,k,l,-i]]

bends=Times@@XingsByArmpits/._[X][a_,b_,c_,d_]:>(Subscript[p, a,-d])(Subscript[p, b,-a])(Subscript[p, c,-b])(Subscript[p, d,-c])

faces=bends//.(Subscript[p, x__,y_])(Subscript[p, y_,z__]):>(Subscript[p, x,y,z])

A=Table[0,Length@faces,Length@faces];
A//MatrixForm

x=XingsByArmpits[[1]]

faces

is=Position[faces,#][[1,1]]& /@ List@@x

Do[is=Position[faces,#][[1,1]]& /@ List@@x;
A[[is,is]]+=If[Head[x]===(SubPlus[X]),
(({{-r, -t, 2 t, t\[Conjugate]}, {-t\[Conjugate], 0, t\[Conjugate], 0}, {2 t\[Conjugate], t, -r, -t\[Conjugate]}, {t, 0, -t, 0}})),(({{r, -t, -2 t\[Conjugate], t\[Conjugate]}, {-t\[Conjugate], 0, t\[Conjugate], 0}, {-2t, t, r, -t\[Conjugate]}, {t, 0, -t, 0}}))],
{x,XingsByArmpits}]

Simplify[A/2/.Conjugate[ω]->(OverBar[ω])]//MatrixForm

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

Plot[Kas[Knot[8,2],(E^(I t))],{t,0,2π}]