<< KnotTheory`

T = X[6, 4, 1, 3] X[4, 2, 5, 1] X[2, 6, 3, 5]

SetAttributes[p,  Orderless]

o1 = 
      T  /.  X[a_, b_, c_, d_]  :>  
          B* p[a, d] p[b, c] + A*p[a, b] p[c, d]

o2 = Expand[o1]

o3 = 
      o2  //.  p[a_, b_] p[b_, c_]  :>  p[a, c]

o4  =  o3  /.  p[_, _]^2  ->  d

o5  =  
      Expand[o4  /.  {B -> 1/A,  
            d ->  -A^2 - 1/A^2}]

PD[Knot[3, 1]]

Jones[%][q]

l = Jones[#][q] &  /@  AllKnots[];

Length[l]

Length[Union[l]]

PD[Knot["K16n230212"]]

Jones[%][q]

l1 = HOMFLYPT[#][a, q] &  /@  AllKnots[];

Length[Union[l1]]