Clear[λ];
K=Knot[8,2]

t=1-(λ^2); r=t+t\[Conjugate];

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

soup=Times@@aps/.(Xp|Xm)[i_,j_,k_,l_]:>a[i,-l]a[j,-i]a[k,-j]a[l,-k]

cs=aps/._[X][i_,j_,k_,l_]:>a[i,-l]a[j,-i]a[k,-j]a[l,-k]//.a[i_,x___,j_]a[j_,y___,k_]:>a[i,x,j,y,k]

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

Do[
is=Position[cs,#][[1,1]]& /@ List@@x;
A[[is,is]]+=If[Head[x]===Xp,
(({{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,aps}
]

A

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

Bed[K_,λ_]:=Module[{t,r,aps,cs,a,A,is},
t=1-(λ^2); r=t+t\[Conjugate];
aps=Times@@PD[K]/.x:X[i_,j_,k_,l_]:>If[PositiveQ[x],(SubPlus[X])[j,k,-l,-i],(SubMinus[X])[-j,k,l,-i]];
cs=aps/._[X][i_,j_,k_,l_]:>a[i,-l]a[j,-i]a[k,-j]a[l,-k]//.a[i_,x___,j_]a[j_,y___,k_]:>a[i,x,j,y,k];
A=Table[0,Length@cs,Length@cs];
Do[is=Position[cs,#][[1,1]]& /@ List@@x;
A[[is,is]]+=If[Head[x]===(SubPlus[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@@aps}];
MatrixSignature[A]
];

{(SubPlus[X]),(SubMinus[X])}//FullForm

a.b//FullForm