Pensieve header: Analysis of k=2 invariants in QU: Aiming to find the relationship with OP2.

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

OverbayP2Data=Get["C:\\drorbn\\AcademicPensieve\\People\\Overbay\\OverbayP2Data.m"];
OP2[K_Knot]:=K/.OverbayP2Data/.T->(T^(1/2));

{Length[OverbayP2Data],Last[OverbayP2Data]}

$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]);

Clear[QP,ω];
QP[Knot[n_,k_]] := QP[Knot[n,k]]=Collect[Module[{fname},
fname="../SL2Invariant/k=2/Data/"<>ToString[n]<>"_"<>ToString[k]<>".m";
Collect[𝔼3[Get[fname][[2,2]]][[3]] // Normal, ϵ, Simplify]
], ϵ, CF];
ω[K_Knot]:=ω[K]=Factor[(((QP@K/.ϵ->0))^(-1))];
(Subscript[c, k_,d_])[K_Knot]:=Factor[SeriesCoefficient[QP[K],{y,0,0}, {ϵ,0,k},{a,0,d}]((ω[K])^(1+2k-d))]

H[p_]:=If[TrueQ@Simplify[p==(p/.T->1/T)],σ@@CoefficientList[Expand@Together[p]/.(T^(n_))/;n<0:>0,T], p];
H[p_]:=If[TrueQ@Simplify[p==(p/.T->1/T)],Style[Expand@Together[p]/.(T^(n_))/;n<0:>0, Background -> Yellow], p];
H[p_]:=If[TrueQ@Simplify[p==(p/.T->1/T)],Style[p, Background -> Yellow], p];
H[l_List]:=H/@l;

MatrixForm[Table[
H/@Factor/@{ω[K],(Subscript[c, 0,0])[K],(Subscript[c, 1,0])[K],(Subscript[c, 1,1])[K],(Subscript[c, 2,0])[K],(Subscript[c, 2,1])[K],(Subscript[c, 2,2])[K]OP2[K]},
{K,AllKnots[{3,7}]}
]]

p1[K_Knot]:=p1[K]=Factor[((T(-(Subscript[c, 1,0])[K]+ω[K]T(∂_T)ω[K]))/(((T-1))^2))];
ρ1[K_Knot]:=ρ1[K]=Factor[((((T-1))^2)/T)p1[K]];

MyCollect[ℰ_,vs_List]:=MyCollect[ℰ,vs,Identity];
MyCollect[ℰ_,vs_List,simp_]:=Total[CoefficientRules[ℰ,vs]/.((ps_->c_):>simp[c]Times@@((vs^ps)))]

Monitor[Total[Table[
Simplify[((Subscript[c, 0,0])[K]==1)∧(2T(∂_T)ω[K]== (Subscript[c, 1,1])[K])∧((Subscript[c, 2,1])[K]==((2 (-1+T) p1[K] ((1+T) ω[K]-3 (-1+T) T(∂_T)ω[K]))/T)+2 ω[K] ((((-1+T))^2) (∂_T)p1[K]+2 (T^2) ((((∂_T)ω[K]))^2)-T ω[K] ((∂_T)ω[K]+T(∂_(T,T))ω[K])))∧((Subscript[c, 2,2])[K]==-2 T (-2 T ((((∂_T)ω[K]))^2)+ω[K] ((∂_T)ω[K]+T (∂_(T,T))ω[K])))],
{K,AllKnots[{3,10}]}]],
K]

MatrixForm[Table[
H/@Together/@Expand /@ {ω[K],p1[K],ρ1[K],-2(Subscript[c, 2,0])[K]+ω[K](Subscript[c, 2,1])[K],OP2[K], OP2[K]/ω[K]},
{K,AllKnots[{3,8}]}
]]

MatrixForm[Table[
Together/@Expand /@ {ω[K],(∂_T)ω[K],p1[K],ρ1[K],-2(Subscript[c, 2,0])[K]+ω[K](Subscript[c, 2,1])[K],OP2[K], OP2[K]/ω[K]},
{K,AllKnots[{3,8}]}
]/.T->1]

MatrixForm[mat=Table[
Factor@{
OP2[K],OP2[K]/.T->-T,(Subscript[c, 2,1])[K],(Subscript[c, 2,2])[K],-2(Subscript[c, 2,0])[K]+ω[K](Subscript[c, 2,1])[K],
((ω[K])^2)ρ1[K],ω[K]ρ1[K],ρ1[K],ω[K]ρ1[K]T(∂_T)ω[K],ρ1[K]T(∂_T)ω[K],T(∂_T)ρ1[K],ω[K]T(∂_T)ρ1[K],((ω[K])^2)T(∂_T)ρ1[K],(T^2)(∂_(T,T))ρ1[K],ω[K](T^2)(∂_(T,T))ρ1[K],(T^2)(∂_(T,T))(ω[K]ρ1[K]),
((ρ1[K])^2),ρ1[K]T(∂_T)ρ1[K],
((ω[K])^4),((ω[K])^3),((ω[K])^2),ω[K],
((ω[K])^3)T(∂_T)ω[K],((ω[K])^2)T(∂_T)ω[K],ω[K] T(∂_T)ω[K], T(∂_T)ω[K],
((ω[K])^2)(((T(∂_T)ω[K]))^2),ω[K](((T(∂_T)ω[K]))^2),(((T(∂_T)ω[K]))^2),
((ω[K])^3) (T^2)(∂_(T,T))ω[K],((ω[K])^2) (T^2)(∂_(T,T))ω[K],ω[K] (T^2)(∂_(T,T))ω[K],(T^2)(∂_(T,T))ω[K],
ω[K](T^3)(∂_(T,T,T))ω[K],(T^3)(∂_(T,T,T))ω[K]
},
{K,AllKnots[{3,8}]}
]];
Dimensions[mat]
MatrixRank[mat/.T->(10^6)]
NullSpace[mat/.T->(10^12)]

MatrixForm[mat=Table[
Factor@{
OP2[K],(Subscript[c, 2,1])[K],(Subscript[c, 2,2])[K],-2(Subscript[c, 2,0])[K]+ω[K](Subscript[c, 2,1])[K],
ω[K]ρ1[K],ρ1[K],ρ1[K]T(∂_T)ω[K],ω[K]T(∂_T)ρ1[K],
((ω[K])^2)T(∂_T)ω[K],ω[K] T(∂_T)ω[K], T(∂_T)ω[K],
ω[K](((T(∂_T)ω[K]))^2),(((T(∂_T)ω[K]))^2),
((ω[K])^2) (T^2)(∂_(T,T))ω[K],ω[K] (T^2)(∂_(T,T))ω[K],
1
},
{K,AllKnots[{8,8}]}
]];
Dimensions[mat]
MatrixRank[mat/.T->(10^4)]
NullSpace[mat/.T->(10^4)]//MatrixForm

Length[AllKnots[{8,8}]]

MatrixForm[mat=Table[
Factor@{
OP2[K],(Subscript[c, 2,1])[K],(Subscript[c, 2,2])[K],-2(Subscript[c, 2,0])[K]+ω[K](Subscript[c, 2,1])[K],
((ω[K])^2)ρ1[K],ω[K]ρ1[K],ρ1[K],ω[K]ρ1[K]T(∂_T)ω[K],ρ1[K]T(∂_T)ω[K],T(∂_T)ρ1[K],ω[K]T(∂_T)ρ1[K],((ω[K])^2)T(∂_T)ρ1[K],(T^2)(∂_(T,T))ρ1[K],ω[K](T^2)(∂_(T,T))ρ1[K],(T^2)(∂_(T,T))(ω[K]ρ1[K]),
((ρ1[K])^2),ρ1[K]T(∂_T)ρ1[K],
((ω[K])^4),((ω[K])^3),((ω[K])^2),ω[K],1,
((ω[K])^3)T(∂_T)ω[K],((ω[K])^2)T(∂_T)ω[K],ω[K] T(∂_T)ω[K], T(∂_T)ω[K],
((ω[K])^2)(((T(∂_T)ω[K]))^2),ω[K](((T(∂_T)ω[K]))^2),(((T(∂_T)ω[K]))^2),
((ω[K])^3) (T^2)(∂_(T,T))ω[K],((ω[K])^2) (T^2)(∂_(T,T))ω[K],ω[K] (T^2)(∂_(T,T))ω[K],(T^2)(∂_(T,T))ω[K]
},
{K,AllKnots[{3,8}]}
]];
Dimensions[mat]
MatrixRank[mat/.T->(10^4)]
NullSpace[mat/.T->(10^4)]//MatrixForm

MatrixForm@H[mat=Table[
Factor@{
-2(Subscript[c, 2,0])[K]+ω[K](Subscript[c, 2,1])[K],
((ω[K])^2)ρ1[K],ω[K]ρ1[K],ρ1[K],(T-(T^(-1)))ω[K]ρ1[K]T(∂_T)ω[K],(T-(T^(-1)))ρ1[K]T(∂_T)ω[K],(T-(T^(-1)))T(∂_T)ρ1[K],(T-(T^(-1)))ω[K]T(∂_T)ρ1[K],(T-(T^(-1)))((ω[K])^2)T(∂_T)ρ1[K],T(∂_T)(T(∂_T)ρ1[K]),ω[K]T(∂_T)(T(∂_T)ρ1[K]),T(∂_T)(T(∂_T)(ω[K]ρ1[K])),
((ρ1[K])^2),ρ1[K](T-(T^(-1)))T(∂_T)ρ1[K],
((ω[K])^4),((ω[K])^3),((ω[K])^2),ω[K],
(T-(T^(-1)))((ω[K])^3)T(∂_T)ω[K],(T-(T^(-1)))((ω[K])^2)T(∂_T)ω[K],(T-(T^(-1)))ω[K] T(∂_T)ω[K], (T-(T^(-1)))T(∂_T)ω[K],
((ω[K])^2)(((T(∂_T)ω[K]))^2),ω[K](((T(∂_T)ω[K]))^2),(((T(∂_T)ω[K]))^2),
((ω[K])^3) T(∂_T)(T(∂_T)ω[K]),((ω[K])^2)T(∂_T)(T(∂_T)ω[K]),ω[K] T(∂_T)(T(∂_T)ω[K]),T(∂_T)(T(∂_T)ω[K]),
(T-(T^(-1)))ω[K]T(∂_T)(T(∂_T)(T(∂_T)ω[K])),(T-(T^(-1)))T(∂_T)(T(∂_T)(T(∂_T)ω[K]))
},
{K,AllKnots[{3,9}]}
]];
Dimensions[mat]
MatrixRank[mat/.T->(10^4)]
NullSpace[mat/.T->(10^4)]//MatrixForm

(mat0=Map[Coefficient[#,T,0]& ,mat, {2}])//MatrixForm

s=(10^5);
(red=LatticeReduce[Prepend[mat0,s IdentityMatrix[Dimensions[mat0][[2]]][[1]]]\[Transpose]]\[Transpose]);
rel=LinearSolve[mat0,Rest[red][[All,Position[red[[1]],s][[1,1]]]]]
Factor@Expand[mat.rel] // Column