Pensieve header: Ad hoc unitarity for one-up one-down pure tangles.

<< KnotTheory`

SetDirectory["C:/drorbn/AcademicPensieve/2015-07/"];
Get["../Projects/MetaCalculi/MetaCalculi-Program.m"]

Ωa=(({{(1/(1-(Subscript[T, 1]))), 0}, {1, (1/(1-(Subscript[T, 2])))}})); Ωb=(({{(1/(1-(Subscript[T, 1]))), 1}, {0, (1/(1-(Subscript[T, 2])))}}));

{L=Knot[10,12], d=2};
n=Crossings[L];
λ=Times@@PD[L] /. X[i_,j_,k_,l_] :> If[PositiveQ[X[i,j,k,l]],Xp[l,i], Xm[j,i]];
ζ=Γ[Drop[λ,{d}]];
{min,max}=Sort[List@@λ[[d]]];
Do[ζ=dm[min+1,i,min+1][ζ], {i,min+2,max-1}];
If[max<2n,
Do[ζ=dm[max+1,i,max+1][ζ], {i,max+2,2n}];
Do[ζ=dm[max+1,i,max+1][ζ], {i, 1, min-1}];
ζ=dσ[min+1->1, max+1->2][ζ],
(*else*) Do[ζ=dm[1,i,1][ζ], {i, 2, min-1}];
ζ=dσ[min+1->2][ζ]
];

λ

U=Simplify[ζ@A]

{Eigenvectors[U], Eigenvalues[U], Simplify[#(#/.(Subscript[T, i_]):>1/(Subscript[T, i]))]& /@ Eigenvalues[U]}

Ωa.Inverse[U] // Simplify // MatrixForm

Transpose[U /. {(Subscript[T, i_]):>1/(Subscript[T, i])}].Ωa // Simplify // MatrixForm

TestUnitarity[L_,d_] := Module[{n,λ,ζ,min,max},
n=Crossings[L];
λ=Times@@PD[L] /. X[i_,j_,k_,l_] :> If[PositiveQ[X[i,j,k,l]],Xp[l,i], Xm[j,i]];
ζ=Γ[Drop[λ,{d}]];
{min,max}=Sort[List@@λ[[d]]];
Do[ζ=dm[min+1,i,min+1][ζ], {i,min+2,max-1}];
If[max<2n,
Do[ζ=dm[max+1,i,max+1][ζ], {i,max+2,2n}];
Do[ζ=dm[max+1,i,max+1][ζ], {i, 1, min-1}];
ζ=dσ[min+1->1, max+1->2][ζ],
(*else*) Do[ζ=dm[1,i,1][ζ], {i, 2, min-1}];
ζ=dσ[min+1->2][ζ]
];
U=Simplify[ζ@A];
Simplify[(Ωa == Transpose[U /. {(Subscript[T, i_]):>1/(Subscript[T, i])}].Ωa.U)∨(Ωb == Transpose[U /. {(Subscript[T, i_]):>1/(Subscript[T, i])}].Ωb.U)]
]

TestUnitarity[L,1]

Table[
Table[TestUnitarity[L,d], {d,Crossings[L]}],
{L,AllKnots[{3,8}]}
]

(U=Simplify[(({{1/α, β/α}, {-γ/α, (α δ-γ β)/α}}))/.{α->1-γ, δ->1-β}]) // MatrixForm

(U=Simplify[(({{1-γ, β}, {γ, 1-β}}))]) // MatrixForm

Factor[(-1+(Subscript[T, 1])) (-1+(Subscript[T, 2]))(Ωa- Transpose[U /. {β->(Subscript[β, c]),γ->(Subscript[γ, c])}].Ωa.U)] // MatrixForm

Eigenvalues[(({{1-γ, β}, {γ, 1-β}}))]

Eigenvalues[(({{(1/(1-γ)), (β/(1-γ))}, {(γ/(-1+γ)), ((-1+β+γ)/(-1+γ))}}))]