# Low Dimensional Topology

## March 31, 2009

### Surgery diagrams from double branched covers

Filed under: 3-manifolds,Knot theory — Jesse Johnson @ 10:00 am

Here’s a neat trick I learned a few weeks ago from Jonathan Bloom at Columbia University.  Let’s say you have a description of a 3-manifold $M$ as the double branched cover of a link $L$ in the 3-sphere, but you would like a description of M as a Dehn surgery on a link in the 3-sphere.  (This sort of thing comes up, for example when comparing different types of Floer and Khovanov homologies.)  If $L$ were the unknot then $M$ would be the 3-sphere and everything is easy.  If $L$ is more complicated than the unknot, then things are a bit more difficult, but here’s what we can do:

At a crossing in a diagram for $L$, we can attach a short arc $\alpha$ that goes from the bottom arc of the crossing to the top arc.  In the double branched cover over $L$, the arc $\alpha$ lifts to a simple closed curve, i.e. a knot $K$ in $M$.  We can simplify $L$ by resolving the crossing at $\alpha$, i.e. replacing the crossing arcs with a pair of arcs whose projections are disjoint.  If $L'$ is the resulting knot in the 3-sphere then the double branched cover $M'$ of $L'$ will be the restult of a Dehn surgery in $M$ along $K$.  (This is often called the Montesinos trick and it has lots of applications in knot theory.)  We can now choose a crossing in $L'$ to get a knot in $M'$, then resolve the crossing and so on, until we have turned $L$ into the unknot.  The double branched cover of the unknot is the 3-sphere, so we have found a sequence of Dehn surgeries that turn $M$ into the 3-sphere.  Going the other way, this gives us a link in the 3-sphere such that Dehn surgery on that link produces $M$.

In fact, the map from the 3-sphere to itself via the double branched cover is simple enough, that the link producing $M$ is fairly easy to construct.  Jonathan has a systematic way of constructing this link, which is described in his recent preprint [1].  (Examples are shown in Figures 3 and 4 of the paper.)  He uses the construction to show that odd Khovanov homology and the Ozsvath-Szabo spectral sequence are mutation invariant.

1. This construction generalizes to give a surgery presentation of any cyclic covering space of S^3 branched over any link, as described for example in Rolfsen’s “Knots and Links”, Chapter 6D. It’s actually quite old- I don’t know exactly how old, but Levine used it in the 1965 paper “A characterization of knot polynomials”.
I’ll take the opportunity to advertise my joint preprint with Andrew Kricker in which we do the same thing for irregular branched dihedral covering spaces (over knots):
http://front.math.ucdavis.edu/0805.2307
(we’ll put up a new version quite soon, so it might be worth waiting a week or two if one wants to download it). I haven’t really got it written up yet, but a similar (slightly souped-up) argument works also for irregular branch coverings associated to finite metacyclic groups and to A_4 (more general metabelian groups are more difficult, and I have no idea how to proceed for non-metabelian permutation groups).
There’s a construction for ANY branched covering space (associated to a representation from the link group onto a finite permutation group) due to Swenton, but it forgets the branch covering (the covering link of the knot):
F.J. Swenton, “Algorithmic construction of Kirby diagrams for branched
covers”, J. Knot Theory Ramifications 13(7) (2004),
pages 939–945.

Comment by Daniel Moskovich — March 31, 2009 @ 8:05 pm

2. Also, I should mention Garoufalidis-Kricker, http://arxiv.org/abs/math/0205328 where the boundary link case is discussed. One of the important things in applications is to have a Kirby theorem “downstairs” which allows you to relate surgery presentations “upstairs”. This allows you to check invariance downstairs of something defined using a surgery presetation “upstairs”, and is the basis for the Kricker-Garoufalidis proof of Rozansky’s rationality conjecture for boundary links.
And on the subject of shameless self-promotion, rather than waiting for us to update on ArXiv, the newer version of our preprint is at:
http://www.sumamathematica.com/JointPaper-2.pdf

Comment by Daniel Moskovich — March 31, 2009 @ 8:28 pm

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