Pensieve header: Converting between PDs and GCs, for Lou Kauffman.

Dear Dror,
Do you have a Gauss Code -----> PD Code
translator in Mathematica that works for virtual knots?
Best,
Lou

Lou -

“Virtual Knots” are honest knots minus error checking. So the silliest conversion program would do - all you have to do is NOT check for planarity.

Best,

Dror.

<< KnotTheory`

GC[K_] := GC @@ (
PD[K] /. X[i_,j_,k_,l_] :> If[PositiveQ[X[i,j,k,l]],
Ar[l,i,+1], Ar[j,i,-1]
]
);

PD[gc_GC] := (PD@@gc) /. {
Ar[i_,j_, +1] :> X[j,i+1,j+1,i],
Ar[i_,j_, -1] :> X[j,i,j+1,i+1]
} /. 2Length[gc]+1 -> 1;

pd = PD[Knot[8,17]]

gc=GC[pd]

PD[gc]

PD[gc]==pd