<< KnotTheory`

K = DTCode[16,8,20,14,2,6,18,10,4,12]

J=Jones[K][q]

Select[AllKnots[],
Jones[#][q] == J &
]

Select[AllKnots[],
Jones[#][q] == (J /. q -> 1/q)&
]

L=PD[
X[1,13,2,22], X[3,14,4,15], X[5,8,6,9], X[7,18,8,19],
X[9,17,10,16], X[11,2,12,3], X[13,7,14,12], X[15,20,16,21],
X[17,6,18,1], X[19,5,20,4], X[21,10,22,11]
]

DrawMorseLink[L]

j=Jones[L][q]

maybe=Select[AllLinks[], (
quo=Factor[(Jones[#][q]/j)];
quo==1 || quo==q || MatchQ[quo, q^_]
)&]

Ooops, silly. L has 11 crossings so it wouldn't be in the tables...