Pensieve header: $A$ - $A^T$ experiments.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\APAI"];

Once[<<KnotTheory`; <<Rot.m];

(Subscript[R, 1])[s_,i_,j_]:=s((Subscript[g, j,i])((Subscript[g, (SuperPlus[j]),j])+(Subscript[g, j,(SuperPlus[j])])-(Subscript[g, i,j]))-(Subscript[g, i,i])((Subscript[g, j,(SuperPlus[j])])-1)-1/2);
ρ[K_]:=ρ[K]=Module[{Cs,φ,n,A,s,i,j,k,Δ,G,ρ1},
{Cs,φ}=Rot[K]; n=Length[Cs];
A=IdentityMatrix[2n+1];
Cases[Cs,{s_,i_,j_}:>(A[[{i,j},{i+1,j+1}]]+=(({{-(T^s), (T^s)-1}, {0, -1}})))];
Δ=(T^((-Total[φ]-Total[Cs[[All,1]]])/2))Det[A];
G=Inverse[A];
ρ1=(∑_(k=1)^n)(Subscript[R, 1])@@Cs[[k]] - (∑_(k=1)^(2n))φ[[k]]((Subscript[g, k,k])-1/2);
Factor@{Δ,(Δ^2)ρ1/.(SuperPlus[α_]):>α+1/.(Subscript[g, α_,β_]):>G[[α,β]]}];

CompareMatrices[A_,B_]:=Grid[
MapThread[Column@*List,{A,B} /. 0->"",2],
Frame->All,ItemSize->All
]

CompareMatrices[IdentityMatrix[2],Table[1,2,2]]

Table[Hue[k/16],{k,0,16}]

AAT[K_] := Module[{Cs,φ,n,A,α,β,k,s,i,j},
{Cs,φ}=Rot[K]; n=Length[Cs];
A=IdentityMatrix[2n+1];
For[k=1,k<=n,k++,
{s,i,j}=Cs[[k]];
A[[{i,j},{i+1,j+1}]]+=Map[Style[#,Background ->Hue[(k/(2n))]]&,(({{-(T^s), (T^s)-1}, {"0", -1}})),{2}]
];
CompareMatrices[A,A\[Transpose]]
]

AAT[Knot[6,3]]