A few posts back, I mentioned the fact that when looking at the double branched cover of a knot, a crossing change downstairs corresponds to an integral Dehn surgery in the cover. Someone pointed out to me that this fact is (part of) what’s called the Montesinos trick. More generally, the Montesinos trick relates Dehn surgeries in the double cover to rational tangle replacements in the original link, and I thought I’d try to explain how this works. Ken Baker put some very nice pictures of this construction on his blog about a year and a half ago. If any of you readers have a favorite application or would like to suggest another source that explains the Montesinos trick, please send in those comments. (Also, Daniel Moskovich mentioned some generalizations of this constructions in comments on the previous post.)
Let be a knot in the 3-sphere, let be an arc in the 3-sphere with endpoints in and interior disjoint from and let be the double branched cover of the 3-sphere, branched along . The branched cover is defined by a projection map that is one-to-one on and two-to-one on the complement of . In particular, it is one-to-one on the endpoints of and two-to-one on the interior, so the preimage in of the arc is a knot in .
A regular neighborhood of in is a ball , whose boundary intersects in four points (two at each endpoint of ). The double branched cover of restricts to a double branched cover of the sphere , so the preimage in of is a torus , which is the boundary of a regular neighborhood of the knot . There is an order two automorphism (i.e. an involution) that fixes the points of that are sent into and interchanges each pair of points that have the same image under . We will use this involution in a minute.
Dehn surgery on along consists of removing this tubular neighborhood of and regluing it by a different gluing map. The homeomorphism type of the resulting 3-manifold is determined by the isotopy type of the gluing map. It happens that any map from to itself can be composed with an isotopy so that the resulting gluing map commutes with the involution , which means that any pair of points in that get sent to the same point in , will be sent by to another such pair of points.
This last fact implies that is the lift of some map from to itself; the map sends the preimage of any point to the preimage of some other point , so we have send to . Now, this may seem kind of silly since up to isotopy, there is precisely one (orientation preserving) automorphism from the sphere to itself, so removing and regluing via the map doesn’t seem to do anything. However, contains two arcs of the knot so when we reglue with this new map, the union of these two arcs with the part of outside forms a new knot . If we ignore the knot, the gluing map is (up to isotopy) the same as before, so this new knot still sits in .
Let be the result of Dehn surgery on along . The claim we started with is that is the double branched cover of , branched along , and we can construct the covering map directly: On the closure of each component of the complement of , we have our new covering map agree with the original covering map . Because the new gluing map upstairs is the lift of the new gluing map downstairs, the covering maps on the two components of the complement match up along and we get a continuous covering map from onto with branch set .