<< KnotTheory`

SetAttributes[P, Orderless]

pd=Knot[10,100] // PD

pd1=Delete[pd, {{2},{6}}]

Times@@pd1 /. X[i_,j_,k_,l_] :> P[i,k]P[j,l] //. P[i_,j_]P[j_,l_] :> P[i,l]

pd2=Delete[pd, {{2},{7}}]

Times@@pd2 /. X[i_,j_,k_,l_] :> P[i,k]P[j,l] //. P[i_,j_]P[j_,l_] :> P[i,l]