Low Dimensional Topology

July 27, 2009

The Montesinos Trick

Filed under: 3-manifolds,Knot theory — Jesse Johnson @ 6:22 pm

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 $K$ be a knot in the 3-sphere, let $\alpha$ be an arc in the 3-sphere with endpoints in $K$ and interior disjoint from $K$ and let $M$ be the double branched cover of the 3-sphere, branched along $K$.  The branched cover is defined by a projection map $p : M \rightarrow S^3$ that is one-to-one on $K$ and two-to-one on the complement of $K$.  In particular, it is one-to-one on the endpoints of $\alpha$ and two-to-one on the interior, so the preimage in $p$ of the arc $\alpha$ is a knot $A$ in $M$.

A regular neighborhood of $\alpha$ in $S^3$ is a ball $B$, whose boundary intersects $K$ in four points (two at each endpoint of $\alpha$).  The double branched cover of $S^3$ restricts to a double branched cover of the sphere $\partial B$, so the preimage in $p$ of $\partial B$ is a torus $T \subset M$, which is the boundary of a regular neighborhood of the knot $A$.  There is an order two automorphism (i.e. an involution) $\phi : T \rightarrow T$ that fixes the points of $T$ that are sent into $K$ and interchanges each pair of points that have the same image under $p$.  We will use this involution in a minute.

Dehn surgery on $M$ along $A$ consists of removing this tubular neighborhood of $A$ 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 $T$ to itself can be composed with an isotopy so that the resulting gluing map $g$ commutes with the involution $\phi$, which means that any pair of points in $T$ that get sent to the same point in $\partial B$, will be sent by $g$ to another such pair of points.

This last fact implies that $g$ is the lift of some map $g'$ from $\partial B$ to itself; the map $g$ sends the preimage of any point $x \in \partial B$ to the preimage of some other point $y \in \partial B$, so we have $g'$ send $x$ to $y$.  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 $B$ and regluing via the map $g'$ doesn’t seem to do anything.  However, $B$ contains two arcs of the knot $K$ so when we reglue with this new map, the union of these two arcs with the part of $K$ outside $B$ forms a new knot $K'$.  If we ignore the knot, the gluing map is (up to isotopy) the same as before, so this new knot $K'$ still sits in $S^3$.

Let $M'$ be the result of Dehn surgery on $M$ along $A$.  The claim we started with is that $M'$ is the double branched cover of $S^3$, branched along $K'$, and we can construct the covering map directly: On the closure of each component of the complement of $T \subset M$, we have our new covering map $p'$ agree with the original covering map $p$.  Because the new gluing map $g$ upstairs is the lift of the new gluing  map $g'$ downstairs, the covering maps on the two components of the complement match up along $T$ and we get a continuous covering map from $M'$ onto $S^3$ with branch set $K'$.

1. 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

2. 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

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