Pensieve header: Invariance of k=2 under Qθ.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\SL2Portfolio"];
<< KnotTheory`
<< "SL2PortfolioProgram.m"

$p=5; $k=2; $U=QU;

(#->CF@ Cθ@(Subscript[𝕆, CU])[{y,a,x} ,#]/. {CU->Times, γ|ℏ->1})& /@ {1, a, ϵ x y,(ϵ^2)(x^2)(y^2)}

(#->Simplify@ Qθ@(Subscript[𝕆, QU])[{y,a,x} ,#]/. {QU->Times, γ|ℏ->1})& /@ {1, a, ϵ x y,(ϵ^2)(x^2)(y^2)}

SCθ[p_]:=Collect[ Cθ@(Subscript[𝕆, CU])[{y,a,x} ,p]/. {CU->Times, γ|ℏ->1}, ϵ, Simplify];
SQθ[p_]:=Collect[ Qθ@(Subscript[𝕆, QU])[{y,a,x} ,p]/. {QU->Times, γ|ℏ->1}, ϵ, Simplify];

(#->SQθ[#])& /@ {1, a, ϵ x y,(ϵ^2)(x^2)(y^2)}

(Subscript[𝔼[L_,Q_,P_], $k_]):=𝔼[L,Q,Series[Normal@P,{ϵ,0,$k}]];
(Subscript[(Subscript[𝔼, d_->r_])[L_,Q_,P_], $k_]):=(Subscript[𝔼, d->r])@@(Subscript[𝔼[L,Q,P], $k]);
𝔼3@𝔼[ω_,L_,Q_,Ps_]:=CF/@(Subscript[𝔼[L,(ω^(-1)) Q,(ω^(-1))((((ω^(-4))ϵ))^(-1+Range@Length@Ps)).Ps], $k]);
𝔼4@𝔼[L_,Q_,P_]:=Module[
{ω=((Normal[P])^(-1))/.ϵ->0,Ps=CoefficientList[P,ϵ]},
CF/@𝔼[ω,L,ω Q,(ω^(-3+4Range@Length@Ps))Ps]];
𝔼3@(Subscript[𝔼, sp___])[as___]:=𝔼3@𝔼[as]/.𝔼->(Subscript[𝔼, sp]);
𝔼4@(Subscript[𝔼, sp___])[as___]:=𝔼4@𝔼[as]/.𝔼->(Subscript[𝔼, sp]);

P[Knot[n_,k_]] := P[Knot[n,k]]=Module[{fname},
fname="../SL2Invariant/k=2/Data/"<>ToString[n]<>"_"<>ToString[k]<>".m";
Collect[𝔼3[Get[fname][[2,2]]][[3]] // Normal, ϵ, Simplify]
]

P[Knot[3,1]]

SCθ@P[Knot[3,1]]

SQθ@P[Knot[3,1]]

Test[K_Knot]:=Simplify[P[K]==SQθ[P[K]]]

Test[Knot[3,1]]

Test /@ AllKnots[{3,8}]

(CF@SST[S𝔻@(Subscript[𝕆, QU])[{y,a,x} ,P@#]-Cθ@S𝔻@(Subscript[𝕆, QU])[{y,a,x} ,P@#]]/.γ|ℏ->1)& /@ AllKnots[{3,7}]