Pensieve header: A semi-Seifert formula for the Alexander polynomial. Continues AlexanderAsFxF.nb at pensieve://Projects/Signatures/.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\Semi-Seifert"];
Once[<<KnotTheory`]

SM for Seifert Matrix:

SM[K_]:=Module[{XingsByArmpits,bends,faces,p,A,is,es,keeps},
XingsByArmpits=List@@PD[K]/.x:X[i_,j_,k_,l_]:>If[PositiveQ[x],(SubPlus[X])[-i,j,k,-l],(SubMinus[X])[-j,k,l,-i]];
bends=Times@@XingsByArmpits/._[X][a_,b_,c_,d_]:>(Subscript[p, a,-d])(Subscript[p, b,-a])(Subscript[p, c,-b])(Subscript[p, d,-c]);
faces=bends//.(Subscript[p, x__,y_])(Subscript[p, y_,z__]):>(Subscript[p, x,y,z]);
A=Table[0,Length@faces,Length@faces];
Do[is=Position[faces,#][[1,1]]& /@ List@@x;
A[[is,is]]+=If[Head[x]===(SubPlus[X]),
(({{-1, -1, 2, 0}, {0, 0, 0, 0}, {0, 1, -1, 0}, {1, 0, -1, 0}})),(({{1, -1, 0, 0}, {0, 0, 0, 0}, {-2, 1, 1, 0}, {1, 0, -1, 0}})) 
],
{x,XingsByArmpits}];
A/2]

SM[Knot[3,1]]//MatrixForm

SSA[K_]:=Module[{A,M,keeps},
A=SM[K];
M=Factor[(T^(1/2))A-(T^(-1/2))A\[Transpose]];
keeps=Complement[Range@Length@M,Union@@FirstPosition[1]/@NullSpace[M]];
Factor@Det[M[[keeps,keeps]]]
]
Union@Table[Simplify[SSA[K]/Alexander[K][T]],{K,AllKnots[{3,10}]}]

Plus[(({{-1, -1, 2, 0}, {0, 0, 0, 0}, {0, 1, -1, 0}, {1, 0, -1, 0}})),(({{1, -1, 0, 0}, {0, 0, 0, 0}, {-2, 1, 1, 0}, {1, 0, -1, 0}}))]/2//MatrixForm

Plus[(({{-1, -1, 2, 0}, {0, 0, 0, 0}, {0, 1, -1, 0}, {1, 0, -1, 0}})),-(({{1, -1, 0, 0}, {0, 0, 0, 0}, {-2, 1, 1, 0}, {1, 0, -1, 0}}))]/2//MatrixForm

(A|->MatrixForm[A-A\[Transpose]])/@{(({{-1, -1, 2, 0}, {0, 0, 0, 0}, {0, 1, -1, 0}, {1, 0, -1, 0}})),(({{1, -1, 0, 0}, {0, 0, 0, 0}, {-2, 1, 1, 0}, {1, 0, -1, 0}}))}