Cabling

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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["http://katlas.org/w/index.php/title=CableComponent.m&action=raw"];

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

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