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 as the double branched cover of a link 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 were the unknot then would be the 3-sphere and everything is easy. If 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 , we can attach a short arc that goes from the bottom arc of the crossing to the top arc. In the double branched cover over , the arc lifts to a simple closed curve, i.e. a knot in . We can simplify by resolving the crossing at , i.e. replacing the crossing arcs with a pair of arcs whose projections are disjoint. If is the resulting knot in the 3-sphere then the double branched cover of will be the restult of a Dehn surgery in along . (This is often called the Montesinos trick and it has lots of applications in knot theory.) We can now choose a crossing in to get a knot in , then resolve the crossing and so on, until we have turned 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 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 .
In fact, the map from the 3-sphere to itself via the double branched cover is simple enough, that the link producing is fairly easy to construct. Jonathan has a systematic way of constructing this link, which is described in his recent preprint . (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.