Pensieve header: Computing the Kauffman bracket.

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

t1=Expand[K /. X[i_,j_,k_,l_] :> A p[i,l]p[j,k]+B p[i,j]p[k,l]]

t2= t1 //. {p[i_,j_]p[j_, k_]:> p[i,k]}

t3=t2 /. {p[i_,i_] :> d, p[i_,j_]^2 :> d}

t4 =Simplify[t3/d /. {B->1/A, d->(-A^2-1/A^2)}]

<<KnotTheory`

Jones[Knot[3,1]][q]

Ks=AllKnots[{0,10}]

Length[Ks]

Js=Jones[#][q]& /@ Ks

Length[Js]

Length[Union[Js]]