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

<< KnotTheory`

EPD[K_]:=EPD@@(PD[K]/.x:X[i_,j_,k_,l_]:>If[PositiveQ[x],(Subscript[X, l,i]),(Subscript[(OverBar[X]), j,i])])

epd=EPD@Knot[10,165]
n=Length[epd]

αRule=βRule={};
epd/.(Subscript[X, i_,j_])|(Subscript[(OverBar[X]), i_,j_]):>(
AppendTo[αRule,{i+1->i,i->i-1}];AppendTo[βRule,{j+1->j,j->j-1}]
);
{αRule,βRule}

AN[1]=0;
epd/.{(Subscript[X, i_,j_]):>(AN[i+1]:=AN[i]+1; AN[j+1]:=AN[j]-1),
(Subscript[(OverBar[X]), i_,j_]):>(AN[i+1]:=AN[i]-1; AN[j+1]:=AN[j]+1)
};
Table[AN[i],{i,2n+1}]

epd/.(Subscript[(x:X|(OverBar[X])), i_,j_]):>(Subscript[x, AN@i,AN@j])