BraidPlot[CollapseBraid@(b=Double@BR[Knot["8_15"]])]

Timing[Homf[b, 7]]

Timing[Homf[b, 7]]

Old Stuff :

Homf[BR[n_, l_], k_] := Expand[Normal[Print["Homf at ", BR[n,l], ", ",k];
Fold[Map[Expand, Dot[#1, HeckeMatrix[n, #2]], {1}]&, SparseArray[{{1} -> 1}, {n!}], l].ClosureMatrix[n,1].ClosureMatrix[n-1,k-1]
]] /. d->(1/v-v)/z // Expand;
Homf[b_BR] := Homf[b, First@b-1];