<< KnotTheory`

ConantNumber[K_, k_] := Times[
SeriesCoefficient[Log[Jones[K][E^x]], {x, 0, k}],
{0,1/3,1/6,4,2,40,60,6720,672,604800,302400,2*13305600,1140480,14529715200,36324288000}[[k]]
];
ConantSeries[K_] := Table[
ConantNumber[K, k],
{k,2,15}
];

ConantSeries[Knot[7,1]]

ConantSeries /@ AllKnots[{3,7}] // MatrixForm

GCD @@@ Transpose[ConantSeries /@ AllKnots[{3,7}]]

Mod[ConantNumber[#, 3]-ConantNumber[#, 15], 13]& /@ AllKnots[]