DataSet2[K_] := DataSet2[K] = (
b=BR[K];
Print[(ds={
Knot -> K,
BraidIndex -> First[b],
H1 -> Homf[b],
H2 -> Homf[db=Double[b]],
H2t -> Homf[db /. {xs__} :> {xs, 1}],
Y2 -> FastY2[b]
}) /. t[i_] :> (Subscript[t, i])];
ds
)

Ks=Select[AllKnots[{3,11}], (BraidIndex[#] == 2)&]

(Flatten[{H1, H2, H2t, Y2} /. (DataSet2[#])]& /@ Ks) /. {z->Random[], v-> Random[], t[1]-> Random[], t[2] -> Random[]}

MatrixRank[%]