Pensieve header: The Alexander polynomial using bridges, tunnels, and Alexander numbering.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2023-12"];
Once[
<< KnotTheory`;
<< "../Talks/Oaxaca-2210/Rot.m"
]

K1=EPD[(Subscript[X, 1,2])];
K3=EPD[(Subscript[X, 1,4]),(Subscript[X, 5,2]),(Subscript[X, 3,6])];
K8=Knot[8,17];
K10=Knot[10,165];

mat[K_,flip_]:=Module[{},
{Cs,φ}=Rot[K]/.{s_Integer,i_,j_}/;flip:>{s,j,i};
n=Length[Cs]; bridges=tunnels={}; bn=tn=0; (* completed features *)
cfb=¬flip; (* current feautre is bridge *); cfs=0; (* current feature start *)
tνut=0;(* Total ν at up transitions *) tνdt=0;(* Total ν at down transitions *)
 ν={lν=0}; ϕ={}; Q=0;
For[k=1,k<=2n,++k,
Cs/.{
{s_,k,j_}:>(
AppendTo[ϕ,bn+1];
If[¬cfb,AppendTo[tunnels,{cfs,k}]; ++tn;cfb=True;cfs=k; tνut+=lν; Q+=(T^(lν))(T-1)(Subscript[β, bn+1])(Subscript[τ, tn])];
AppendTo[ν,lν+=s];
),
{s_,i_,k}:>(
AppendTo[ϕ,tn+1];
If[cfb,AppendTo[bridges,{cfs,k}]; ++bn;cfb=False;cfs=k; tνdt+=lν; Q+=(T^(lν))(1-T)(Subscript[β, bn])(Subscript[τ, tn+1]) ];
AppendTo[ν,lν-=s];
)
};
If[k==1,ffb=cfb];
Cs/.{
{s_,k,j_}/;k>j:>(Q+=(T^(ν[[k]]))(T-1)((T^s)-1)(Subscript[β, ϕ[[k]]])(Subscript[τ, ϕ[[j]]])),
{s_,i_,k}/;k>i:>(Q+=(T^(ν[[i]]))(T-1)((T^s)-1)(Subscript[β, ϕ[[i]]])(Subscript[τ, ϕ[[k]]]))
}
];
Factor@Table[(((∂_((Subscript[β, i]),(Subscript[τ, j])))Q)/(T-1)),{i,Max[bn,tn]},{j,Max[bn,tn]}]
];
mat[K_]:=mat[K,False];
mat[Flip@K_]:=mat[K,True];

K=K3

mat[K]//MatrixForm

mat[Flip@K]//MatrixForm

Factor[mat[K]+(mat[Flip@K]\[Transpose]/.T->(T^(-1)))]

Select[AllKnots[{3,11}],Union@Flatten@Factor[mat[#]+(mat[Flip@#]\[Transpose]/.T->(T^(-1)))]=!={0}&]

Reverse@Sort[Reverse/@Tally[
Union@Table[A=mat[K];
u=Factor[(((T^(-Total[φ]/2))((((T^(-1/2))))^(tνut+tνdt))Det[A])/(Alexander[K][T]))(((-1))^(Last@ϕ))If[cfb,-1,1]];{(*Total[First/@Cs],*)(*Total[ν],*)(*Total[Cs/.{s_Integer,i_,j_}:>s ν[[i]]],*) 
u},
{K,AllKnots[{3,11}]}],
Most[#1]===Most[#2]&]]

Total@Table[A=mat[K];Simplify[(((-1))^(Last@ϕ))If[cfb,-1,1]Alexander[K][T]==(T^(-Total[φ]/2))((((T^(-1/2))))^(tνut+tνdt))Det[A]],{K,AllKnots[{3,11}]}]

Total@Table[A=mat[K];Total[First/@Cs]==-tνut+tνdt,{K,AllKnots[{3,10}]}]