CableComponent.m

CableComponent[BR[n_Integer, js_List], K_] :=
 Module[{BC, C0, C1, C2, CC1, CS1, CS2, L, S, a, e, h, i, i1, i2, j, 
   j1, j2, jss, k, k1, kjs, out, out0, out1, p, p1, pos, q, r, s, ss,
    t, t0, t1, t2, tj, v, w, writhe},
  L = PD[K];
   kjs = BR[L]2;
  For[i1 = 1; writhe = 0, i1 <= Length[kjs], i1++, 
   writhe = writhe + Sign[kjsi1]];
   For[i2 = 1; jss[0] = js, i2 <= n Abs[writhe], i2++, 
   jss[i2] = 
    Flatten[{jss[i2 - 1], Table[-Sign[writhe] e, {e, n - 1}]}]];
  k1 = Length[jss[n Abs[writhe]]];
   For[i = 1, i <= n, i++, s[i] = a[i] = i];
  For[
   j = 1, j <= k1, j++,
   p = Select[Range[n], Abs[jss[n Abs[writhe]]j] == a[#] &][[
     1]];
   q = Select[Range[n], a[#] == a[p] + 1 &]1;
   If[jss[n Abs[writhe]]j > 0, 
    K[j] = X[s[q], n + 2 j, n + 2 j - 1, s[p]], 
    K[j] = X[s[p], s[q], n + 2 j, n + 2 j - 1]];
   s[p] = n + 2 j;
   s[q] = n + 2 j - 1;
   a[p]++;
   a[q]--
   ];
  BC = Table[K[d], {d, k1}];
  If[Jones[L][q] === 1,
   For[j1 = 1, j1 <= Length[BC], j1++, 
    For[i = 1, i <= n, i++, BCj1 = BCj1 /. s[i] :> a[i]
     ]];
   If[BC == {}, BC = {Loop[1]}];
   out1 = PD @@ BC,
   For[j2 = 1, j2 <= Length[BC], j2++, 
    For[tj = 1, tj <= n, tj++, BCj2 = BCj2 /. tj :> 1[tj]]
    ];
   p1 = Select[Range[n], # != s[#] &];
   S = Select[L, MemberQ[#, 1] && MemberQ[#, 2] & ];
   pos = Position[S, 1]1, 2;
   r = Select[Table[i, {i, Length[L]}], L# == Flatten @@ S &][[
     1]];
   k = 0;
   out0 = L /. X[a_, b_, c_, d_] :> (
       ++k;
       Table[
          X[h[i, j - 1, k], v[i, j, k], h[i, j, k], v[i - 1, j, k]],
          {i, n}, {j, n}
          ] /. {h[i_, 0, _] :> a[i], h[i_, n, _] :> c[i]} /. If[
         d - b == 1 || b - d > 1,
         {v[0, j_, _] :> d[j], 
          v[n, j_, _] :> b[j]}, {v[0, j_, _] :> d[n + 1 - j], 
          v[n, j_, _] :> b[n + 1 - j]}
         ]
       );
   w = Flatten@out0r;
   out = PD @@ Flatten[Join @@ out0];
   ss = Table[a[i], {i, n}]p1;
    CC1 = List @@ out;
   For[t0 = 1, t0 <= Length[ss], t0++, 
    C0[t0] = Select[w, MemberQ[#, 1[sst0]] &]; 
    C1[t0] = 
     Select[C0[t0], 
      Mod[Position[#, 1[sst0]]1, 1, 2] == Mod[pos, 2] &]; 
    C2[t0] = 
     C1[t0] /. 1[sst0] :> 
       s[Select[Range[n], a[#] == sst0 &]1]];
   
   CS1 = Flatten[Table[C1[t1], {t1, Length[ss]}]];
    CS2 = Flatten[Table[C2[t2], {t2, Length[ss]}]];
   For[i = 1, i <= Length[CS1], i++, 
    CC1 = DeleteCases[CC1, CS1i]];
   out1 = Union[BC, CC1, CS2];
   PD @@ out1;
   k = 0;
   out1 = 
    PD @@ ( out1 /. ((# -> ++k) & /@ (List @@ Union @@ out1)))]];