<< KnotTheory`

K=PD[
X[1,13,2,12],X[3,14,4,15],X[5,9,6,8],X[7,18,8,19],
X[9,21,10,20],X[11,22,12,23],X[13,17,14,16],X[15,2,16,3],
X[17,5,18,4],X[19,6,20,7],X[21,1,22,24],X[23,10,24,11]
];
J=Jones[K][q]

Select[AllKnots[12], Jones[#][q]==J&]

Select[AllKnots[12], Jones[#][q]==(J/.q->1/q)&]

So the above knot appears to be (Subscript[12, 503]^a).