Pensieve Header: Testing TailScatteringTowardsTheHead.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2010-09"];
<< KnotTheory`
<< TailScatteringTowardsTheHead-Program.m

Union[
Simplify[
Det[Drop[R[GD[#]], None, -1]] / Alexander[#][X]
]& /@ AllKnots[{3,10}]
]

Test[K_] := (K -> Simplify /@ Expand /@ Equal[
EZ[GD[K]] ,
- x X D[Alexander[K][X], X] / Alexander[K][X] + (SL[K] /. DL -> x+DR)
])

Test /@ AllKnots[8]

Test2[K_] := (K -> Simplify /@ Expand /@ Equal[
EZ[GD[K]]  /. DR -> 0,
EZ2[GD[K]]
])

Test2 /@ AllKnots[8]

{K=Knot[10,162], A=Alexander[K][X], gd=GD[K], n=Length[gd],arcs=Arcs[gd], Draw[gd]}

ez2=EZ2[gd]

Simplify[Expand[Plus@@EZ1[gd] /. DR -> 0]]

R[gd]  // MatrixForm

MatrixForm /@ {M=Drop[R[gd], None, -1], DM = X D[M, X]}

EA = Simplify[Expand[-x Tr[Inverse[M].DM]]]

IM=Append[Inverse[Drop[R[gd], None, -1]], Table[0, {n}]];
j=0;
Simplify[Expand[-x  (Plus @@ (gd /. Ar[t_, h_, s_] :>
(++j; (-IM[[arcs[[t]], j]] + IM[[arcs[[h]]+1, j]])s X^s)
))]]

IM= Inverse[Append[R[gd], e[n+1, n+1]]];
j=0;
Simplify[Expand[-x  (Plus @@ (gd /. Ar[t_, h_, s_] :>
(++j; (-IM[[arcs[[t]], j]] + IM[[arcs[[h]]+1, j]])s X^s)
))]]

Simplify[EA-ez2]

j=0;
Simplify[Expand[-x  (List @@ (gd /. Ar[t_, h_, s_] :>
(++j; (-IM[[arcs[[t]], j]] + IM[[arcs[[h]]+1, j]])s X^s)
))]]

Simplify[Expand[EZ1[gd] /. DR -> 0]]

Det[Drop[R[gd],None, {#}]]& /@ Range[Length[gd]]

EZ[gd]

EZ1[gd]

Simplify[x X D[Log[Alexander[K][X]], X]]

SL[K]

EZ[gd] /. DR -> 0

n=Length[gd]; R[gd] // MatrixForm

r=R[gd]; r[[All, 1]] += r[[All, n+1]]; r = Drop[r, None, -1];
MatrixForm[r]

Det[r]

Det[Drop[ r, -1, -1]]

Alexander[K][X]

Union @@ Minors[r]

Union[Expand[X^(-Exponent[#, X, Min])#]& /@ Union @@ Minors[r]]

PolynomialGCD @@ Union[Expand[X^(-Exponent[#, X, Min])#]& /@ Union @@ Minors[r]]