Cabling

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CableComponent[BR[n,js],K] returns the $n$-th cable of the knot $K$ with the braid on n strands with crossings js = {j1, j2, ...} inserted in it. It also performs the necessary number of 1/n-twists on the components of the cable to compensate for a non-zero writhe number of the original knot. Cabling knot 3_1, for instance, and inserting the braid BR[3,{1,2}], we get:

(For In[1] see Setup)

In[2]:=	CableComponent[BR[3, {1, 2}], Knot[3, 1]] // DrawMorseLink
Out[2]=	-Graphics-

For some special cases, we can check our result using Burau's Theorem.

In[3]:=

CableComponent[n_integer, K_] := CableComponent[BR[n, {}], K]; 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[kjs[[i1]]]]; 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++, BC[[j1]] = BC[[j1]] /. 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++, BC[[j2]] = BC[[j2]] /. 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@out0[[r]]; 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[ss[[t0]]]] &]; C1[t0] = Select[C0[t0], Mod[Position[#, 1[ss[[t0]]]][[1, 1]], 2] == Mod[pos, 2] &]; C2[t0] = C1[t0] /. 1[ss[[t0]]] :> s[Select[Range[n], a[#] == ss[[t0]] &][[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, CS1[[i]]]]; out1 = Union[BC, CC1, CS2]; PD @@ out1; k = 0; out1 = PD @@ ( out1 /. ((# -> ++k) & /@ (List @@ Union @@ out1)))]];