The Program

<< KnotTheory`

wAlex[gc_GC] := {
MatrixForm[
AA=DiagonalMatrix[List @@ gc /. Ar[t_,h_,s_] :> s]
];
Bij[Ar[ti_,hi_,si_], Ar[tj_,hj_,sj_]] := If[
ti<hj<hi || hi<hj<ti,
Sign[hi-ti]*Sign[hj-tj]*(X^(-sj*Sign[hj-tj])-1),
0
];
MatrixForm[BB=Outer[Bij,List@@gc, List@@gc]];
Cij[Ar[ti_,hi_,si_], Ar[tj_,hj_,sj_]] := If[
ti≤hj≤hi || hi≤hj≤ti,
-Sign[hi-ti]*x,
0
];
MatrixForm[CC=Outer[Cij,List@@gc, List@@gc]];
MatrixForm[
DD=AA.Inverse[IdentityMatrix[Length[BB]]-BB].BB.CC
 //Together // ExpandNumerator
];
Tr[DD] //Together // ExpandNumerator // ExpandDenominator
};
wAlex[K_] := Join[{
pd=PD[K],
gc=GC @@ pd /. X[i_,j_,k_,l_] :> If[PositiveQ[X[i,j,k,l]],
Ar[l,i,+1], Ar[j,i,-1]
]
},
wAlex[gc]
];
wAlexander[K_] := wAlexander[K] = Last[wAlex[K]];
wAlex2[gc_GC] := {
Bij[Ar[ti_,hi_,si_], Ar[tj_,hj_,sj_]] := If[
ti<hj<hi || hi<hj<ti,
X^(-sj*Sign[hj-tj])-1,
0
];
MatrixForm[BB=Outer[Bij,List@@gc, List@@gc]],
Cij[Ar[ti_,hi_,si_], Ar[tj_,hj_,sj_]] := If[
ti≤hj≤hi || hi≤hj≤ti,
-Sign[hj-tj]*sj*x,
0
];
MatrixForm[CC=Outer[Cij,List@@gc, List@@gc]],
MatrixForm[
DD=Inverse[IdentityMatrix[Length[BB]]-BB].BB.CC
 //Together // ExpandNumerator
],
Tr[DD] //Together // ExpandNumerator // ExpandDenominator
};
wAlex2[K_] := Join[{
pd=PD[K],
gc=GC @@ pd /. X[i_,j_,k_,l_] :> If[PositiveQ[X[i,j,k,l]],
Ar[l,i,+1], Ar[j,i,-1]
]
},
wAlex2[gc]
];
wAlexander2[K_] := Last[wAlex2[K]];

And @@ (
(wAlexander[#]==wAlexander2[#])& /@ AllKnots[{3,9}]
)

(# -> wAlex2[#])& /@ AllKnots[{3,7}]

{
Length[Ks=AllKnots[{3,12}]],
Length[Union[Alexander[#][t]& /@ Ks]],
Length[Union[wAlexander[#]& /@ Ks]],
Length[Union[Pair[Alexander[#][t], wAlexander[#]]& /@ Ks]]
}

Ks=AllKnots[10];
As= Alexander[#][t]& /@ Ks;
Ps=Position[As, #][[1,1]]& /@ As;
Position[MapThread[Equal,{Ps, Range[Length[Ps]]}], False]

Ks=AllKnots[10];
As= wAlexander[#]& /@ Ks;
Ps=Position[As, #][[1,1]]& /@ As;
Position[MapThread[Equal,{Ps, Range[Length[Ps]]}], False]

Alexander[#][t]& /@ {Knot[10,42], Knot[10,75]}

wAlexander /@ {Knot[10,42], Knot[10,75]}

test[K_] := (
wAlexander[K] == ExpandDenominator[ExpandNumerator[Factor[
-x*X*D[Alexander[K][X], X]/Alexander[K][X]
]]]
)

test[Knot[10,75]]

And @@(test /@ AllKnots[{3,11}])

And @@(test /@ AllKnots[12, Alternating])

And @@(test /@ AllKnots[12, NonAlternating])

The smallest former breakdown :

wAlex[GC[Ar[5,3,-1], Ar[6,2,+1]]]

wAlex[GC[Ar[1,4,+1],Ar[5,3,-1], Ar[6,2,+1]]]

A long 4 crossing positive knot is a long trefoil

Compare[common_GC, extras_List] := Thread[
wAlex[common~Join~#]& /@ extras
];

Compare[GC[Ar[1,8,+1]],
{
GC[Ar[7,2,+1], Ar[12,3,+1], Ar[6,11,+1]],
GC[Ar[6,3,+1], Ar[11,2,+1], Ar[7,12,+1]]
}
]

wAlex[GC[Ar[1,6,+1], Ar[5,2,+1], Ar[8,3,+1], Ar[4,7,+1]]]

wAlex[GC[Ar[1,8,+1], Ar[7,2,+1], Ar[12,3,+1], Ar[4,9,+1], Ar[5,10,-1], Ar[6,11,+1]]]

wAlex[GC[Ar[1,8,+1], Ar[11,2,+1], Ar[6,3,+1], Ar[4,9,+1], Ar[5,10,-1], Ar[7,12,+1]]]

Other checks :

wAlex[GC[Ar[1,3,+1],Ar[4,2,+1]]]

wAlex[GC[Ar[1,5,+1],Ar[6,4,-1], Ar[7,3,+1], Ar[8,2,+1]]]

wAlex[BR[2,{1}]]

wAlex[BR[2,{1,1,-1}]]

wAlex[BR[2,{1,1,1}]]

wAlex[BR[3,{1,2,1,2}]]

wAlex[BR[3,{-1,2,-1,2}]]

wAlex[BR[3,{1,-2,1,-2}]]

wAlex[BR[3,{-2,1,-2,1}]]

BraidPlot[BR[3,{1,2,1,1,-1,2}]]

wAlex[BR[3,{-1,2,1,2,1,1}]]

wAlex[BR[3,{1,2,1,1,-1,2}]]

Table[
{k, wAlex[
GC[Ar[2,5,1],Ar[3,10,1],Ar[6,11,1],Ar[12,7,1],Ar[1,8,-1],Ar[9,4,1]] /. Ar[t_,h_,s_] :> Ar[(k+t)~Mod~12, (k+h)~Mod~12, s]
]},
{k,5,6}
]

wAlex[BR[3,{1,2,1,2,-2,2}]]

wAlex[Knot[4,1]]

wAlex[Mirror[Knot[4,1]]]

wAlex[Knot[5,1]]

wAlex[Knot[10,132]]

wAlex[BR[2,{-1,-1,-1}]]

Morals

wAlex2[Knot[10,132]]

BB=(({{0, 0, 0, 0, 0, 0, 0, 0, 0, -1+X}, {-1+(1/X), 0, -1+(1/X), 0, 0, 0, 0, 0, 0, 0}, {0, -1+(1/X), 0, 0, -1+(1/X), 0, 0, 0, -1+X, 0}, {0, 0, 0, 0, 0, -1+X, 0, 0, 0, 0}, {0, 0, 0, -1+(1/X), 0, 0, -1+(1/X), 0, -1+X, 0}, {0, 0, 0, -1+(1/X), 0, 0, -1+(1/X), 0, -1+X, 0}, {0, 0, 0, -1+(1/X), 0, -1+X, 0, -1+X, 0, 0}, {0, 0, 0, -1+(1/X), 0, -1+X, 0, 0, 0, 0}, {0, -1+(1/X), 0, 0, -1+(1/X), 0, 0, 0, 0, 0}, {-1+(1/X), 0, -1+(1/X), 0, 0, 0, 0, 0, 0, 0}})); CC=(({{-x, 0, 0, 0, 0, 0, 0, 0, 0, x}, {-x, -x, -x, 0, 0, 0, 0, 0, 0, 0}, {0, -x, -x, 0, -x, 0, 0, 0, x, 0}, {0, 0, 0, -x, 0, x, 0, 0, 0, 0}, {0, 0, 0, -x, -x, 0, -x, 0, x, 0}, {0, 0, 0, -x, 0, x, -x, 0, x, 0}, {0, 0, 0, -x, 0, x, -x, x, 0, 0}, {0, 0, 0, -x, 0, x, 0, x, 0, 0}, {0, -x, 0, 0, -x, 0, 0, 0, x, 0}, {-x, 0, -x, 0, 0, 0, 0, 0, 0, x}}));

BB.CC// Simplify

{CC/x, -IdentityMatrix[Length[BB]]+(D[BB, X] /. X->1)}

{1,-1}.%

% // MatrixForm

The example on http : // katlas.math.toronto.edu/drorbn/bbs/show?shot = Archibald - 080709 - 110259. jpg is :

wAlex2[GC[Ar[1,3,+1], Ar[2,5,-1], Ar[6,4,+1]]]