Pensieve header:  Comparing $G$ and $G^T$.

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

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

DepthList[K_]:=Module[{Cs,n,dd,k,s,i,j},
Cs=Rot[K][[1]]; n=Length[Cs];
dd=Table[0,2n+1];
For[k=1,k<=n,k++,{s,i,j}=Cs[[k]];dd[[{i+1,j+1}]]={s,-s}];
FoldList[Plus,dd]
]

DepthList/@ AllKnots[{3,7}]

K=Knot[4,1];
A:= Module[{Cs,φ,n,A,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}]]+=(({{-(T^s), (T^s)-1}, {0, -1}}))];
A
];
G:=Inverse[A];
DM:=DiagonalMatrix[(T^(DepthList[K]))]; DM
A//MatrixForm
G//MatrixForm

Factor[A.G]//MatrixForm

Factor[G.(DM/.T->(T^1)).(A\[Transpose]/.T->(T^1))]//MatrixForm