<< KnotTheory`

Alexander[Knot[11,NonAlternating,185]][t]

NumberOfKnots[11,NonAlternating]

A knot for which the (Subscript[ρ, 1]) genus bound is right but the Alexander genus bound is wrong is (Subscript[K, 293]^n):

K=Knot[12,NonAlternating,293]

PD[K]

DrawMorseLink[K]