Pensieve header: Analysis of k=2 invariants in QU.

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

OverbayP2Data=Get["C:\\drorbn\\AcademicPensieve\\People\\Overbay\\OverbayP2Data.m"];

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

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];

(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]
];
QP[K_Knot]:=QP[K]=CF@P[K];

MatrixForm[AllKnots[{3,7}] /. K_Knot :> {K, qp=Collect[QP@K/.{y->0,a->-1/2},{ϵ,a},Factor]; ω=(((qp/.ϵ->0))^(-1)),P1=Expand[(*((T )/(((T-1))^2))*)(ω^3)Coefficient[qp,ϵ]],P2=Expand[(ω^5)Coefficient[qp,(ϵ^2)]],(*Factor[ ((P2-(P2/.T->-T))/(4(T+1/T)))],*)Expand[2P2+(T+1/T) ω P1],OP2=K/.OverbayP2Data/.T->(T^(1/2)),{ω,(∂_T)ω,P1,P2,OP2}/.T->1}]

MatrixForm[mat=AllKnots[{3,8}] /. K_Knot :> {K, qp=Collect[QP@K/.{y->0},{ϵ,a},Factor]; ω=(((qp/.ϵ->0))^(-1)),P1=Expand[(*((T )/(((T-1))^2))*)(ω^3)Coefficient[qp,ϵ]/.a->0],P2=Expand@Factor[(ω^5)Coefficient[qp,(ϵ^2)]/.a->-1/2], OP2=K/.OverbayP2Data/.T->(T^(1/2)),{ω,T(∂_T)ω,T(∂_T)(T(∂_T)ω),P1,P2,OP2}/.T->-1}]

MatrixForm[mat=AllKnots[{3,8}] /. K_Knot :> (K; qp=Collect[QP@K/.{y->0},{ϵ,a},Factor]; ω=(((qp/.ϵ->0))^(-1));P1=Expand[(*((T )/(((T-1))^2))*)(ω^3)Coefficient[qp,ϵ]/.a->0];P2=Expand@Factor[(ω^5)Coefficient[qp,(ϵ^2)]/.a->-1/2];OP2=K/.OverbayP2Data/.T->(T^(1/2));
{q1=ω,q2=T(∂_T)(T(∂_T)ω),q3=P1,q4=T(∂_T)P1,q5=T(∂_T)(T(∂_T)P1),q1 q2, q1 q3, q2 q3,OP2,P2}/.T->-2)]
Dimensions[mat]
MatrixRank[mat]

NullSpace[mat]

MatrixForm[mat=AllKnots[{3,8}] /. K_Knot :> (K; qp=Collect[QP@K/.{y->0},{ϵ,a},Factor]; ω=(((qp/.ϵ->0))^(-1));P1=Expand[(*((T )/(((T-1))^2))*)(ω^3)Coefficient[qp,ϵ]/.a->0];P2=Expand@Factor[(ω^5)Coefficient[qp,(ϵ^2)]/.a->-1/2];OP2=K/.OverbayP2Data/.T->(T^(1/2));
{q1=ω,q2=T(∂_T)(T(∂_T)ω),q3=P1,q4=T(∂_T)P1,q1 q2, q1 q3, q2 q3,OP2,P2}/.T->-2)]
Dimensions[mat]
MatrixRank[mat]