(* ::Package:: *)

(* ::Text:: *)
(*Pensieve Header: An improved MVA program with Jana Archibald.*)


(* ::Input:: *)
(*<< KnotTheory`*)


(* ::Input:: *)
(*MultivariableAlexander2[PD[Loop[_]]]:=(1/(#[1]-1))&*)
(*MultivariableAlexander2[K_]/;Head[K]=!=PD:=MultivariableAlexander2[PD[K]]*)
(**)
(*MultivariableAlexander2[pd_PD]:=MultivariableAlexander2[pd]= Module[*)
(*{l, mat, skel, pd1, G, t, arcs, path, i,j,k, M, emb, done, pd2, rot, place},*)
(*l=Length[pd];*)
(*mat=Table[0, {2*l}, {2*l}];*)
(*skel=Skeleton[pd];*)
(*pd1=List@@pd;*)
(*G=\!\(\**)
(*TagBox[*)
(*RowBox[{"Table", "[", *)
(*RowBox[{"0", ",", *)
(*RowBox[{"{", *)
(*RowBox[{"2", "*", " ", "l"}], "}"}], ",", *)
(*RowBox[{"{", "l", "}"}]}], "]"}],*)
(*Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\);*)
(*pd1//.X[a_, b_, c_, d_]:> If[d==b+1||b-d>1,*)
(*{mat[[c,a]] =-t[b]; mat[[c,b]]=t[a]-1;  mat[[c,c]]=1},*)
(*{mat[[c,a]]=-1;mat[[c,b]]=1-t[a] ; mat[[c,c]]=t[b]}*)
(*];*)
(*arcs=Times@@pd/.{*)
(*X[i_,j_,k_,l_]/;(l-j==1||j-l>1):>path[k]path[i] path[j,l],*)
(*X[i_,j_,k_,l_]/;(j-l==1||l-j>1):>path[k]path[i] path[l,j],*)
(*P[i_,j_]:>path[i,j]*)
(*}//.{*)
(*path[a__,i_]path[i_,b__]:>path[a,i,b],*)
(*path[a__,i_]path[b__,i_]:>Join[path[a,i],Reverse[path[b]]],*)
(*path[i_,a__]path[i_,b__]:>Join[Reverse[path[b]],path[i,a]],*)
(*path[a__,i_]path[i_]:>path[a,i],*)
(*path[i_,a__]path[i_]:>path[a,i],*)
(*path[i_]path[i_]:>path[i]*)
(*};*)
(*If[Length[arcs]===l,For[i=1,i<=2*l,i++,*)
(*G=ReplacePart[G,1,{i,First[First[Position[arcs,i]]]}]*)
(*]];*)
(*mat=mat/. t[a_]:> t[Position[skel,a][[1,1]]];*)
(*If[Length[arcs]===l,*)
(*M=Factor[Simplify[*)
(*Det[*)
(*Delete[*)
(*Transpose[Delete[*)
(*Transpose[G].mat.G,*)
(*Position[arcs,pd1[[1,3]]][[1,1]]*)
(*]],*)
(*Position[arcs,pd1[[1,3]]][[1,1]]*)
(*]*)
(*]/( t[ Position[skel,pd1[[1,3]]][[1,1]]]-1)*)
(*]],*)
(*M=0];*)
(*emb=Table[Null,{Length[pd]}];*)
(*done=Table[Null, {2*Length[pd]}];*)
(*emb[[1]]=0;*)
(*pd2=pd;*)
(*rot=Table[0, {Length[skel]}];*)
(*place[i_, a_] := Module[*)
(*{ni, na, arc, dir, oparc},*)
(*arc=pd2[[i,a]];*)
(*{{ni, na}}=Complement[Position[pd2,arc], {{i,a}}];*)
(*If[emb[[ni]]===Null,*)
(*emb[[ni]]=3-a+emb[[i]];*)
(*pd2[[ni]]=RotateLeft[pd1[[ni]], na-1];*)
(*place[ni, #]& /@ {2,3,4},*)
(*(* Else *)  oparc=RotateLeft[pd2[[i]], 2][[a]];*)
(*If[done[[arc]]===Null,*)
(*done[[arc]]=1;*)
(*dir=If[arc-oparc==1 || arc-oparc<-1, 1, -1];*)
(*rot[[Position[skel, arc][[1,1]]]] += dir*(emb[[ni]]-emb[[i]]+a-na-2)*)
(*]*)
(*]*)
(*];*)
(*place[1,#]& /@ {1,2,3,4};*)
(*k=-rot/4;*)
(*For[j=1,j<=l,j++,*)
(*k=ReplacePart[k,-1+k[[Position[skel,pd[[j,2]]][[1,1]]]],Position[skel,pd[[j,2]]][[1,1]]]*)
(*];*)
(*For[i=1,i<=Length[k],i++,*)
(*M*=t[i]^((1/2)*k[[i]])*)
(*];*)
(*If[pd[[1,4]]==pd[[1,2]]+1||pd[[1,2]]-pd[[1,4]]>1,*)
(*M*=t[Position[skel,pd[[1,1]]][[1,1]]]*t[Position[skel,pd[[1,2]]][[1,1]]],*)
(*M*=t[Position[skel,pd[[1,1]]][[1,1]]]*)
(*];*)
(*Evaluate[M/.t->#]&*)
(*]*)


(* ::Input:: *)
(*MV=MultivariableAlexander; MV2=MultivariableAlexander2*)


(* ::Input:: *)
(*test1[L_] := ( *)
(*mv=MV[L][t]; mv2=First[MV2[L][t]];*)
(*Or @@ Map[*)
(*(mv1=mv /. t[i_] :> t[#[[i]]]; Head[Expand[Simplify[mv2/mv1]]]=!=Plus)&,*)
(*Permutations[Range[Length[Skeleton[L]]]]*)
(*]*)
(*);*)
(*Print[# ->test1[#]]& /@ AllLinks[9];*)


(* ::Input:: *)
(*{MV[Link[9,NonAlternating,27]][t], MV2[Link[9,NonAlternating,27]][t]}*)


(* ::Input:: *)
(*Flip[X[i_,j_,k_,l_]] := If[l==j+1 || j-l>1, X[j,k,l,i], X[l,i,j,k]];*)
(*VCube[pd_, l_List] := Module[*)
(*{f},*)
(*Expand[pd*Times @@ ((1-f[#])& /@ l)] //. pd1_PD*f[i_] :> MapAt[Flip, pd1, i]*)
(*]*)


(* ::Input:: *)
(*Series[VCube[PD[#], {1,2, 7}] /. pd_PD :> Jones[pd][E^x], {x, 0,3}]& /@AllLinks[8]*)


(* ::Input:: *)
(*Print[# -> Series[VCube[PD[#], {1,2, 7}] /. pd_PD :> MV2[pd][t] /. t[i_] -> E^(h x[i]), {h, 0, 2}]]& /@AllLinks[8];*)


(* ::Input:: *)
(*Print[# -> Series[VCube[PD[#], {1,2, 7}] /. pd_PD :> MV2[pd][t] /. t[i_] -> E^(h x[i]), {h, 0,2}]]& /@AllLinks[9];*)


(* ::Input:: *)
(*Print[# -> Series[VCube[PD[#], {1,2, 7,8}] /. pd_PD :> MV2[pd][t] /. t[i_] -> E^(h x[i]), {h, 0,2}]]& /@AllLinks[9];*)


(* ::Input:: *)
(*Print[# -> Series[VCube[PD[#], {1,2,5, 7,9}] /. pd_PD :> MV2[pd][t] /. t[i_] -> E^(h x[i]), {h, 0, 3}]]& /@AllLinks[11];*)