Pensieve Header: The separation powers of various invariants.

Loading the invariants

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\Theta"];

Do[
NumberOfKnots[n,Alternating]={0,0,1,1,2,3,7,18,41,123,367,1288,4878,19536,85263}[[n]];
NumberOfKnots[n,NonAlternating]={0,0,0,0,0,0,0,3,8,42,185,888,5110,27436,168030}[[n]],
{n,15}
];
NumberOfKnots[n_Integer]:=NumberOfKnots[n,Alternating]+NumberOfKnots[n,NonAlternating];
NumberOfKnots[{n_}]:=Sum[NumberOfKnots[n1],{n1,n}];
{NumberOfKnots[15,Alternating], NumberOfKnots[15,NonAlternating],NumberOfKnots[15],NumberOfKnots[{15}]}

The data files for H, Kh, and Θ are available at https://drorbn.net/AcademicPensieve/Projects/HigherRank/Data.

Get["../../Projects/HigherRank/Data/HOMFLYPT3-15.m"]/.(K_->V_):>(H[K]=Expand@V);
H[Knot[3,1]]

Get["../../Projects/HigherRank/Data/Kh3-15.m"]/.(K_->V_):>(Kh[K]=Expand@V);
Kh[Knot[3,1]]

J[K_]:=J[K]=Kh[K]/.t->-1;
J[Knot[3,1]]

Get["../../Projects/HigherRank/Data/theta3-15.m"]/.(K_->V_):>(Θ[K]=Expand@V);
Θ[Knot[3,1]]

Δ[K_]:=Δ[K]=Θ[K][[1]];
(Subscript[ρ, 1])[K_]:=(Subscript[ρ, 1])[K]=-Θ[K][[2]]/.{T1->T, T2->1};
Factor@{Δ[Knot[3,1]], (Subscript[ρ, 1])[Knot[3,1]]}

The data files for (Subscript[ρ, 2]) are available at https://drorbn.net/AcademicPensieve/Talks/KnotTheoryCongress-2502/Rho2Data.

(Join@@(Get["../../Talks/KnotTheoryCongress-2502/Rho2Data/"<>#]& /@ {"Rho2_3-10.m","Rho2_11.m","Rho2_12.m","Rho2_13.m","Rho2_14.m","Rho2_15.m"}))/.(K_->V_):>((Subscript[ρ, 2])[K]=Expand@V);
(Subscript[ρ, 2])[Knot[3,1]]

The data file for Vol is available at https://drorbn.net/AcademicPensieve/Projects/Theta/Vols/.

Get["../../Projects/Theta/Vols/Vols.m"]/.(K_->v_):>(Vol[K]=v);
Vol[Knot[4,1]]
Vol[Knot[4,1]]//Precision

LowVolKnots=Select[AllKnots[{3,15}], Vol[#]<Vol[Knot[4,1]]&]

Vol/@LowVolKnots

Warning: σ was evaluated only up to 14 crossings!

fs=FileNames["../../Talks/ICERM-2305/Data/*.m"];
ls=Table[f->Length[Get[f]/.(K_->V_):>(σ[K]=V)], {f,fs}]
Total[Last/@ls] (* Expect 59937 *)
σ[Knot[3,1]]
σ[Knot[14,NonAlternating,27436]]

Stats

Knots[n_Integer]:=If[n<=10,
Table[Knot[n,k],{k,NumberOfKnots[n]}],
Join[
Table[Knot[n,Alternating,k],{k,NumberOfKnots[n,Alternating]}],
Table[Knot[n,NonAlternating,k],{k,NumberOfKnots[n,NonAlternating]}]
]
];
Knots[{n_}]:=Join@@Table[Knots[n1],{n1,n}];
Knots[{11}]//Short

Stats[{n_},ιs_List]:=NumberOfKnots[{n}]-CountDistinct@Table[ι[K],{K,Knots[{n}]},{ι,ιs}];
Stats[ιs_List]:=Monitor[
Table[def=Stats[{n},ιs], {n,10,14}],
{n-1,def}
];

Stats[{Δ}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{Δ,σ}]

Stats[{J}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,J}]

Stats[{Kh}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,Kh}]

Stats[{H}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,H}]

Stats[{H,Kh}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,H,Kh}]

Table[d->Stats[{If[Vol@#<1,0,N[Vol@#,d]]&}], {d,0,70}]//Column

Clear[TVol]
TVol[K_]:=TVol[K]=If[Vol@K<1,0,N[Vol@K,58]]

Stats[{TVol}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,TVol}]

Stats[{Kh,H,TVol}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,Kh,H,TVol}]

Stats[{Δ,(Subscript[ρ, 1])}]

Stats[{Δ,(Subscript[ρ, 1]),(Subscript[ρ, 2])}]

Stats[{(Subscript[ρ, 1]),(Subscript[ρ, 2]),Kh,H,TVol}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,(Subscript[ρ, 1]),(Subscript[ρ, 2]),Kh,H,TVol}]

Stats[{Θ}]

6758/41432//N

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,Θ}]

Stats[{Θ,Kh}]

Stats[{Θ,H}]

Stats[{Θ,H,Kh}]

Stats[{Θ,(Subscript[ρ, 2])}]

Stats[{Θ,TVol}]

Stats[{Θ,(Subscript[ρ, 2]),TVol}]

Stats[{Θ,(Subscript[ρ, 2]),Kh}]

Stats[{Θ,(Subscript[ρ, 2]),H}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,Θ,(Subscript[ρ, 2])}]

Stats[{Θ,(Subscript[ρ, 2]),H,Kh}]

Stats[{Θ,(Subscript[ρ, 2]),TVol}]

Stats[{Θ,(Subscript[ρ, 2]),H,Kh,TVol}]

Warning: σ was evaluated only up to 14 crossings!

Stats[{σ,Θ,(Subscript[ρ, 2]),H,Kh,TVol}]