Pensieve header: The two-point function and  the modified two-point function both have simple inverses. (Added Jan 23: A complete blunder - G1 is the same as G here).

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\APerturbedAlexanderInvariant"];
Once[<<KnotTheory`; <<RVK.m];

K=Knot[8,8];
RVK[K]

{Cs,r}=List@@RVK[K]

n=Length[Cs]; B=Table[0,2n,2n+1];
Do[{s,i,j}=c;B[[{i,j},{i,j,i+1,j+1}]]=(({{1, 0, -1, 0}, {0, 1, (T^s)-1, -(T^s)}})),{c,Cs}];

B//MatrixForm

A=B[[All,2;;]];
A//MatrixForm

A1=A;
Do[A1[[i+1,i]]=0,{i, 2n-1}];
A1//MatrixForm

Inverse[A1]//Simplify//MatrixForm

G=Simplify@Prepend[Table[0,2n]][Inverse[A]]; (Subscript[g, α_,β_]):=G[[α,β]];
G//MatrixForm

G1=Simplify@Inverse[A];
Do[G1[[i,i+1]]==1, {i,2n-1}]
G1//Simplify//MatrixForm

Inverse[G1]//Simplify//MatrixForm