Pensieve header: With Andrey Khesin - coincidences with 9_27, 9_41, 11n4, and 11n21.

<<KnotTheory`

Alexander[Knot[9,27]][t]

Alexander[Knot[9,27]][t] // Factor

Alexander[Knot[9,41]][t] // Factor

Import["http://katlas.org/w/index.php?title=IdentifyWithin.m&action=raw"]

IdentifyWithin[Knot[9,41], AllKnots[{9,11}], Invariants->{Jones[#][q]&}]

Expand[Jones[Knot[9,27]][q]]

Expand[Jones[Knot[9,41]][q]]

Position[Expand[Jones[#][q]& /@ AllKnots[{9,11}]], Expand[Jones[Knot[9,41]][q]]]

Position[Expand[Jones[#][q]& /@ AllKnots[{9,11}]], Expand[Jones[Knot[9,41]][1/q]]]

AllKnots[{9,11}][[{585,602}]]

Alexander[#][t]& /@ {Knot[11,NonAlternating,4],Knot[11,NonAlternating,21]}

HOMFLYPT[#][q,t]& /@ {Knot[11,NonAlternating,4],Knot[11,NonAlternating,21]}