Pensieve header: Evaluating ρ1 and ρ2 at $T=1$.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2020-01"];
<< KnotTheory`
$k=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_]:=Simplify/@(Subscript[𝔼[L,(ω^(-1)) Q,(ω^(-1))((((ω^(-4))ϵ))^(-1+Range@Length@Ps)).Ps], $k]);
𝔼3@(Subscript[𝔼, sp___])[as___]:=𝔼3@𝔼[as]/.𝔼->(Subscript[𝔼, sp]);

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

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

p2[K_Knot]:=p2[K]=Expand[-2(Subscript[c, 2,0])[K]+ω[K](Subscript[c, 2,1])[K]];

MatrixForm@(mat=Table[{1,Vassiliev[2][K], Vassiliev[3][K],p1[K]/.T->1}, {K, AllKnots[{3,7}]}])

NullSpace[mat]

Vassiliev[4,1][K_]:=Coefficient[Conway[K][z],(z^4)]

Table[Vassiliev[4,1][K], {K, AllKnots[{3,7}]}]

Vassiliev[4,2][K_]:=SeriesCoefficient[4Jones[K][(E^x)],{x,0,4}]

mat=Table[{Vassiliev[4,1][K],Vassiliev[4,2][K]}, {K, AllKnots[{3,8}]}]

MatrixRank[mat]

MatrixForm@(mat=Table[{1,Vassiliev[2][K], Vassiliev[3][K],((Vassiliev[2][K])^2),Vassiliev[4,1][K],Vassiliev[4,2][K],p1[K]/.T->1,p2[K]/.T->1}, {K, AllKnots[{3,8}]}])

NullSpace[mat]

MatrixForm@(mat=Table[{1,Vassiliev[2][K], Vassiliev[3][K],((Vassiliev[2][K])^2),Vassiliev[4,1][K],Vassiliev[4,2][K], Together[(((∂_T)Alexander[K][T])/(T-1))]/.T->1,Together[(∂_(T,T))Alexander[K][T]]/.T->1, Together[(∂_(T,T,T))Alexander[K][T]]/.T->1}, {K, AllKnots[{3,8}]}])

NullSpace[mat]