Low Dimensional Topology

June 30, 2008

Knot complements covering knot complements

Filed under: 3-manifolds,Knot theory — Ken Baker @ 12:47 pm

I thought I’d follow up on the construction beginning Commensurability and make a post rather than a comment. Gonzalez-Acuna and Whitten actually show that if a knot complement (i.e. the complement of a knot in S^3) finitely covers another knot complement, then the covering is cyclic [1]. This was done algebraically, and now in the post-Perelman world it gives a characterization of knots in S^3 with lens space surgeries through the construction Jesse described. Genevieve Walsh pointed this out to me last summer.
One may find the prospect of using this characterization to approach the Berge Conjecture tantalizing.

Instead of performing a p/1-lens space surgery on a knot followed by taking the p-fold S^3 cover of the resulting lens space to get another knot in S^3, one can do it the other way around. Take a cover then do surgery. More specifically, take the p-fold cyclic branched cover of the knot, then do 1-surgery to obtain S^3 again.

Through Heegaard Floer homology, we now know that such knots must be fibered (see Ni and the references therein). Consequentially we can describe the monodromy of the covering knot in S^3. If a knot with p/1-lens space surgery has monodromy \phi, then the knot with covering complement is a fibered knot of the same genus with monodromy \tau \phi^p (where, \tau is a Dehn twist along the boundary).

Dunno if this approach will really help elucidate a resolution to the Berge conjecture, but it does present the following problem. Given a knot in S^3, you can either (a) take a p-fold cyclic branched cover and then do 1-surgery on the result to obtain manifold A or (b) do p/1-surgery and then take a p-fold cover (a smallest covering in which the knot lifts to a null homologous one perhaps?) to obtain manifold B. To what extent do the manifolds A and B differ?

[1] F. Gonzalez-Acuna and W. C. Whitten. Imbeddings of knot groups in knot groups. Geometry and topology (Athens, Ga., 1985), 147–156, Lecture Notes in Pure and Appl. Math., 105, Dekker, New York, 1987


1 Comment »

  1. Well, because you’re starting in S^3, the answer is rather straightforward by considering the lack of ambiguity in taking a p-fold cyclic covering of a knot complement.

    Maybe a less straightforward problem is to identify the manifolds obtained from those commensurable dodecahedral knots by filling (all the boundary components of) the commensurator along the lifts of the S^3 meridians of these knots.

    (Ack. Here’s the right post.)

    Comment by kennethleebaker — July 5, 2008 @ 1:18 pm | Reply

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