(* ::Package:: *)

(************************************************************************)
(* This file was generated automatically by the Mathematica front end.  *)
(* It contains Initialization cells from a Notebook file, which         *)
(* typically will have the same name as this file except ending in      *)
(* ".nb" instead of ".m".                                               *)
(*                                                                      *)
(* This file is intended to be loaded into the Mathematica kernel using *)
(* the package loading commands Get or Needs.  Doing so is equivalent   *)
(* to using the Evaluate Initialization Cells menu command in the front *)
(* end.                                                                 *)
(*                                                                      *)
(* DO NOT EDIT THIS FILE.  This entire file is regenerated              *)
(* automatically each time the parent Notebook file is saved in the     *)
(* Mathematica front end.  Any changes you make to this file will be    *)
(* overwritten.                                                         *)
(************************************************************************)



(* ::Input::Initialization:: *)
Once[<<KnotTheory`];
Once[<<"../Projects/Profile/Profile.m"]


(* ::Input::Initialization:: *)
RVK::usage="RVK[xs, rots] represents a Rotational Virtual Knot with a list of n Xp/Xm crossings xs and a length 2n list of rotation numbers rots. Crossing sites are indexed 1 through 2n, and rots\[LeftDoubleBracket]k\[RightDoubleBracket] is the rotation between site k-1 and site k. RVK is also a casting operator converting to the RVK presentation from other knot presentations.";
RVK[pd_PD] := Module[{n,xs,x,rots,front,k},
n=Length[pd];
xs=List@@pd /. x_X :> If[PositiveQ[x], Xp[x[[4]],x[[1]]], Xm[x[[2]],x[[1]]]];
rots=Table[0, {2n}];
front={0};
For[k=0, k<2n, ++k,
If[k==0\[Or]FreeQ[front, -k],
front = Flatten[front /. k->Catch[xs /. {
Xp[k+1,l_]|Xm[l_,k+1] :> Throw[{l,k+1,1-l}],
Xp[l_,k+1]|Xm[k+1,l_] :> (++rots[[l]]; Throw[{1-l,k+1,l}])
}]],
If[MatchQ[front,{___,k,___,-k,___}], --rots[[k+1]]]
]
];
RVK[xs,rots]
];
RVK[K_] := RVK[PD[K]];


(* ::Input::Initialization:: *)
Subscript[ul, _]=Subscript[nl, _]=rot[_,0] = \[DoubleStruckCapitalE][1,0,0,0];
rot[i_,1] := Subscript[ur, i];
rot[i_,n_Integer] /; n>1 := Module[{y}, rot[i,n-1]rot[y,1] // Subscript[m, i,y->i]];
rot[i_,-1] := Subscript[nr, i];
rot[i_,n_Integer] /; n<-1 := Module[{y}, rot[i,n+1]rot[y,-1] // Subscript[m, i,y->i]];


(* ::Input::Initialization:: *)
Subscript[t, _]=t;
Z[K_] := Z[RVK@K];
Z[rvk_RVK] := Subscript[PP, z]@Module[{todo,n,rots,\[Zeta],done,st,x, \[Zeta]1, i,j,k,k1,k2,k3},
{todo,rots}=List@@rvk;
AppendTo[rots,0];
n=Length[todo];
\[Zeta]=\[DoubleStruckCapitalE][1,0,0,0];
done={0};
st=Range[0,2n+1];
While[todo=!={},
{x}=MaximalBy[todo, Length[done\[Intersection]{#[[1]], #[[2]], #[[1]]-1,#[[2]]-1}]&,1];
Z$todo=todo; Z$x=x;
{i,j}=List@@x;
\[Zeta]1=Switch[Head[x],
 Xp, Subscript[m, j,k->j][\!\(
\*SubsuperscriptBox[\(R\), \(i, j\), \(+\)] \((\(\(
\*SubsuperscriptBox[\(R\), \(k3, k\), \(-\)] 
\*SubscriptBox[\(nr\), \(k1\)] 
\*SubscriptBox[\(ul\), \(k2\)] // 
\*SubscriptBox[\(m\), \(k, k1 -> k\)]\) // 
\*SubscriptBox[\(m\), \(k, k2 -> k\)]\) // 
\*SubscriptBox[\(m\), \(k, k3 -> k\)])\)\)],
 Xm, Subscript[m, j,k->j][\!\(
\*SubsuperscriptBox[\(R\), \(i, j\), \(-\)] \((\(\(
\*SubsuperscriptBox[\(R\), \(k, k3\), \(+\)] 
\*SubscriptBox[\(nr\), \(k1\)] 
\*SubscriptBox[\(ul\), \(k2\)] // 
\*SubscriptBox[\(m\), \(k, k1 -> k\)]\) // 
\*SubscriptBox[\(m\), \(k, k2 -> k\)]\) // 
\*SubscriptBox[\(m\), \(k, k3 -> k\)])\)\)]
];
\[Zeta]1=rot[k,rots[[i]]]\[Zeta]1 // Subscript[m, k,i->i]; rots[[i]]=0;
\[Zeta]1=\[Zeta]1 rot[k, rots[[i+1]]] // Subscript[m, i,k->i]; rots[[i+1]]=0;
\[Zeta]1=rot[k,rots[[j]]]\[Zeta]1 // Subscript[m, k,j->j]; rots[[j]]=0;
\[Zeta]1=\[Zeta]1 rot[k, rots[[j+1]]] // Subscript[m, j,k->j]; rots[[j+1]]=0;
\[Zeta]*=\[Zeta]1;
If[MemberQ[done, i],  \[Zeta]=\[Zeta]//Subscript[m, i,i+1->i]; st=st/.st[[i+2]]->st[[i+1]]];
If[MemberQ[done, i-1],  \[Zeta]=\[Zeta]//Subscript[m, st[[i]],i->st[[i]]]; st=st/.st[[i+1]]->st[[i]]]; 
If[MemberQ[done, j],  \[Zeta]=\[Zeta]//Subscript[m, j,j+1->j]; st=st/.st[[j+2]]->st[[j+1]]]; 
If[MemberQ[done, j-1],  \[Zeta]=\[Zeta]//Subscript[m, st[[j]],j->st[[j]]]; st=st/.st[[j+1]]->st[[j]]];
done=done\[Union]{i-1,i,j-1,j};
todo=DeleteCases[todo, x]
];
\[Zeta] /. {Subscript[e, 0]->e,Subscript[l, 0]->l,Subscript[f, 0]->f}
]
