Pensieve header:  Analysis of the Kashaev Invariant..

A=(({{v, u, 1, u}, {u, 1, u, 1}, {1, u, v, u}, {u, 1, u, 1}}))/.v->2(u^2)-1

NullSpace[A]

{u,-1}.NullSpace[A]

Once[<< KnotTheory`]

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

Kas[K_,u_]:=Kas[K,u]=Module[{v,XingsByArmpits,bends,faces,p,A,is},
v=2(u^2)-1 ;
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];
Do[is=Position[faces,#][[1,1]]& /@ List@@x;
A[[is,is]]+=If[Head[x]===(SubPlus[X]),
(({{v, u, 1, u}, {u, 1, u, 1}, {1, u, v, u}, {u, 1, u, 1}})),-(({{v, u, 1, u}, {u, 1, u, 1}, {1, u, v, u}, {u, 1, u, 1}}))],
{x,XingsByArmpits}];
A ];

MatrixForm[A=Kas[Knot[3,1],u]]

NullSpace[A]//MatrixForm

Table[NullSpace[Kas[K,u]]//MatrixForm,{K,AllKnots[{3,8}]}]

Table[NullSpace[Kas[K,u]]//Length,{K,AllKnots[{3,12}]}]//Union

Table[NullSpace[Kas[K,u]]//Length,{K,AllKnots[{3,13}]}]//Union

Table[Simplify@NullSpace[Kas[K,u]],{K,AllKnots[{3,13}]}]//Flatten//Union

Table[Simplify@NullSpace[Kas[K,u]],{K,AllKnots[{3,14}]}]//Flatten//Union

Table[NullSpace[(∂_u)Kas[K,u]]//MatrixForm,{K,AllKnots[{3,7}]}]

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]];

Module[{K=(Subscript[8, 2]),XingsByArmpits,bends},
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]);
bends//.(Subscript[p, x__,y_])(Subscript[p, y_,z__]):>(Subscript[p, x,y,z])]

2u NullSpace[Kas[(Subscript[8, 2]),u]]//MatrixForm