Update: Andy Wand has posted a preprint of his proof on the arXiv. Obviously, definitions and content there supercede what was originally written in this series of blog posts.
Update: (12 May 2014) Some minor technical details have been corrected (Thanks to Patrick Massot for pointing these out)
Here is the third and final post on the Legendrian surgery conjecture.
In the previous post, we went through H-K-M’s proof that a contact structure is overtwisted if and only if there is a supporting open book with a
left-veering non-right-veering arc. This gives a criterion for establishing tight vs ovetwisted, but its not invariant under positive stabilization. So we can’t just check one supporting open book for , we have to check every single one, and we have to check every possible arc in each open book to see if it is right veering or not.
But the problem might be somewhat manageable:
Let’s start with a simple example. Suppose our monodromy is not efficient and some arc , starts off heading to the right but then immediately crosses back over , forming a trivial bigon that can be isotoped away. If we remove this bigon by an isotopy, then appears to be
left-veering non-right-veering now.
The arc is right-veering if and only if the curves form a second trivial bigon, allowing us to isotope back across .
So, if we have a bigon formed by the pair , we want to be able to find a second bigon that completes it, meaning that the two cancel each other out and we preserve right-veering-ness.
Now a second example. Suppose we have a square region, as in the following local picture, and that the monodromy is efficient. Then are right-veering. However, suppose that we can destabilize along the right arc. Remember, this means that do not intersect except at the boundary. The destabilization means that we cut the surface along and then add a negative Dehn twist along a closed curve isotopic to . This is the green curve in the second picture.
Notice what happens to after the destabilization: it forms a trivial bigon with . So we can isotope so that it becomes
left-veering non-right-veering. If that was it, then this contact structure would be overtwisted.
However, suppose that there is a second square region as in the following picture, forming a “singular annulus”
Now, when we destabilize, the second square region becomes a trivial bigon that pulls all the way around the diagram to cancel out the bigon formed from the first square region.
So, if a contact structure is tight and if we have a square bounded by such that we can destabilize along one of the arcs, we need to be able to find a second square bounded by the same four arcs and connected at opposite corners. In other words, we need to be able to find a second square that (completes) .
Lets go back to an example from the second post, where we saw how a positive stabilization can hide a
left-veering non-right-veering arc. In that example, we can find such a square bounded by the original arcs and the new arcs added by the stabilization. Furthermore, notice that there is a completing square if and only if crosses back over .
Before we continue, lets fix some terminology. A (region) is an immersed 2n-gon in a page whose boundary lies in the curves , alternating between an and a . Furthermore, let’s orient the curves , subject to the condition that for each i, is a closed, oriented curve. With this orientation, we call a region (positive) if the induced orientation on the boundary agrees with the orientation on the arc basis and a region is (negative) otherwise. Note that an oriented arc basis does not necessarily induce a coherent orientation on the boundary of a region.
Third example: Now let’s take the local picture near a square region and stabilize again. After stabilization, we can find a hexagon region.Note that this region is “completed”, meaning there is a second hexagon region attached at the corners of the first region, if and only if the square region before the stabilization was “completed”.
We can repeat this process by positively stabilizing as many times as we want. The notion of a “completed” region, of any number of sides, persists after repeated positive stabilizations. And if we can destabilize along the various sides of our 2n-gon, we know we can reduce it back to the case of the canceling bigons in the first example above.
We can also look for uncompleted regions further into the surface: Let’s suppose we have a chain of regions as in the following picture:
There are two square regions, connected to the boundary, and we can orient the arc basis so these regions are positive. Then implies that must be negative regions. Finally, is a positive region, that is not connected to the boundary at all. Now let’s do something really tricky: stabilize along the green arc that passes through all five regions.
Afterwards, we get the following picture:
In effect, the three positive regions have been amalgamated into a single octagonal region. The regions have been amalgamated as well and they have also been joined with the negative regions beyond . The result is that the 8-gon region we find after stabilizing is “completed” if and only if was “completed” (meaning there is a square negative region forming a singular annulus, as in example 2 above).
Again, “completion” is a property that persists after positive stabilizations. So, if we have an open book for some contact structure, we can look for uncompleted regions in the interior that are connected to the boundary by a chain of regions; then by a stabilization, bring it to the boundary. Andy refers to any a monodromy with no such uncompleted regions as consistent and it should be a stabilization-invariant characterization of tightness.
These are just some suggestive examples: it seems like a decent amount of work to clean this up into an algorithm that takes an open book with an uncompleted region and, by a sequence of positive
(de)stabilizations, find an open book with a left-veering non-right-veering arc.
Finally, let’s get back to the Legendrian surgery conjecture. Andy never elaborated much on this aspect, leaving it as a corollary of the characterization of tightness in terms of open books.
However, the basic idea is that Legendrian surgery can be realized as adding to the monodromy a positive Dehn twist along some essential curve in some open book supporting . The key step now is to prove that positive Dehn twist preserve the consistency of the monodromy or equivalently, that a positive Dehn twist does not create an uncompleted region.