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 .
Like this:
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The best reference might be:
J. Montesinos, “Three manifolds as 3-fold branched covers of S^3″, Quart. J. Math. Oxford (2), 27 (1976), 85-94.
It’s elegant, but don’t like the Montesinos trick because the fact that the surgery arc intersects the knot loses all information on covering linkage (as far as I can see). The process in Rolfsen’s book (straightforward Dehn surgery) preserves that information.
I really wish I could prove that any 3-manifold arises as a 3-fold branched cover of S^3 by Dehn surgery alone (no arcs)- show that any 3-manifold has a surgery presentation which is a 3-fold lift of a surgery presentation of a knot (in, say, the complement of a (3n,2)-torus knot). I would like this much more than the Montesinos trick- it would be much neater and more powerful in my opinion. I thought about this for a while at one point, but I’m seriously stuck.
Comment by Daniel Moskovich — August 3, 2009 @ 10:36 am |
Another nice entry point to this subject is Lickorish’s paper “Prime Knots and Tangles” from Trans. Amer. Math. Soc. (1981). Also, Eudave-Mu\~noz has made significant use of the Montesinos trick in a number of papers.
Comment by Scott Taylor — August 4, 2009 @ 8:19 am |