Clear[λ];
K=Knot[8,2]

soup=Times@@PD[K]/.x:X[i_,j_,k_,l_]:>If[PositiveQ[x],
(Subscript[a, j,i])[i](Subscript[a, k,-j])[j](Subscript[a, -l,-k])[k](Subscript[a, -i,l])[l],
(Subscript[a, -j,i])[i](Subscript[a, k,j])[j](Subscript[a, l,-k])[k](Subscript[a, -i,-l])[l]
]

cs=soup//.(Subscript[a, i_,j_])[x__](Subscript[a, j_,k_])[y__]:>(Subscript[a, i,k])[x,y]/.(Subscript[a, __])[x__]:>a[x]

A=Table[0,Length@cs,Length@cs]

t=1-(λ^2); r=t+t\[Conjugate];

{t,r}

Do[Echo[x];
is=Position[cs,x[[#]]][[1,1]]&/@{1,2,3,4};
Echo[is];
A[[is,is]]+=If[PositiveQ[x],
(({{0, t\[Conjugate], 0, -t\[Conjugate]}, {t, -r, -t\[Conjugate], 2t\[Conjugate]}, {0, -t, 0, t}, {-t, 2t, t\[Conjugate], -r}})),(({{r, -t, -2 t\[Conjugate], t\[Conjugate]}, {-t\[Conjugate], 0, t\[Conjugate], 0}, {-2t, t, r, -t\[Conjugate]}, {t, 0, -t, 0}}))
],
{x,List@@PD[K]} ]

A

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