# “Reverse Cabling” a link possible?

You can cable any link. Since methinks the Wiki definition allows random twists in the parallel lines, define: a line - doubles = and one crossing + is replaced by four #. (All horizontal lines are hereby declared overpasses.) Call that canonic cable if it has no name yet. Of course, if you started with k link components, you now have 2k. And very neat, cabled Reidemeister 2 and 3 moves are legal! (That's why I insisted on this special cabling.) The annoying part is Reidemeister 1. A kink turns into a double twist. (Cabled R1+ and R1- still annihilate.) But that begs the question: Can any link L with 2k components be "reverse cabled", i.e. a "mother link" l exists with canonic cable(l)=L? The Whitehead link would be the first possible counterexample. (If it is: What is the crucial difference between links that have a "mother" and those who haven't?)

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One crucial difference is the existence of an essential torus in a cable complement. –  Jim Conant Jun 20 '11 at 16:09
Rephrasing Jim's comment: the answer is generally no. But when a link is a cable (which is very rare), there is a canonical de-cabling provided you suitably normalize things. Take a look at the "satellite knot" Wikipedia page. There's several references there where these details are worked out. As-is, this question is maybe more appropriate for math.stackexchange. –  Ryan Budney Jun 20 '11 at 16:34