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CableComponent[BR[n,js],K], whose code is available here, 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]:= Import[""];
In[3]:= CableComponent[BR[3, {1, 2}], Knot[3, 1]] // DrawMorseLink
Cabling Out 3.gif
Out[3]= -Graphics-

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