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)
The Legendrian surgery conjecture is one of the biggest problems in contact topology. The question is, does (-1) Legendrian surgery on Legendrian knots in a contact 3-manifold preserve tightness? Andy Wand has an approach to proving this by characterizing the tightness of some in terms of the monodromy of an open book supporting the contact structure . He gave a seminar talk last month at Georgia Tech that I attended and there is also a video of a similar talk at the Simons Center. He hasn’t posted a written proof yet, an hour-long lecture is not sufficient to cover all the details and I don’t completely understand every part yet, so I can’t verify the proof for myself. However, I do understand some suggestive examples that motivate his approach and I want to blog that. I’ll start by reviewing some background in contact topology, motivating the conjecture and describing Honda-Kazez-Matic’s work on non-right-veering monodromies before moving on to Andy’s work.
Let’s start from the beginning. An overtwisted disk in a contact 3-manifold in an embedded disk such that the contact structure is tangent to the disk along its boundary. For example, consider the unit disk in the contact structure . A contact structure is overtwisted if it contains an overtwisted disk and is tight otherwise.
There is nice classification of overtwisted contact structures, due to Eliashberg.
Theorem. (Eliashberg) Let M be a closed, oriented 3-manifold. Then there is 1-1 correspondence between overtwisted contact structures on M, up to isotopy, and homotopy classes of 2-plane fields on M.
Therefore, the classification of overtwisted contact structures is governed by algebraic topology and is not really interesting from a geometric perspective. Understanding and classifying tight contact structures on 3-manifolds is a big remaining topic in contact topology.
Motivation for Legendrian surgery conjecture
One way to construct contact 3-manifolds is by performing Dehn surgery but in a manner that respects the geometry. Recall that -Dehn surgery on a knot consists of removing a tubular neighborhood of the knot, and then regluing via the diffeomorphism that sends the meridian to , where is a longitude (equivalently a framed pushoff of the knot). This operation is well-defined up to isotopy but depends upon the (framing), the explicit identification of . Denote the new manifold as
In contact topology, a Legendrian knot has a framing, the Thurston-Bennequin framing, that is well-defined up to Legendrian isotopy and neighborhoods of all Legendrian knots are contactomorphic. This framing is given by any vector field in M along K that is positively transverse to the contact planes, such as the Reeb vector field for any contact form. When there is a well-defined null-homologous framing, the Thurston-Bennequin framing is often written the integer giving the difference between the null-homologous framing and the TB framing. To properly keep track of surgery framings, coefficients in parentheses will indicate framing relative to the TB framing and coefficients without parentheses will indicate framing relative to the null-homologous framing.
When the surgery coefficient is for some integer k, the contact structure on the complement extends uniquely across the surgery torus. When this coefficient is (-1), this operation is called Legendrian surgery.
In 3-manifold topology, the Lickorish-Wallace theorem states that every closed, oriented 3-manifold can be obtained by integral surgery on a link in . There is an analogous result in contact topology:
Theorem (Ding-Geiges) Every closed contact 3-manifold is given by -surgery on a Legendrian link in .
Here means that we perform (-1) surgery on some link components and (+1) surgery on the remaining components.
For example, the standard contact structure on is given by (+1) surgery on the unknot. The standard overtwisted contact structure on , which is contactomorphic to the one described above in the definition of overtwisted disks, is given by (+1) surgery on the Hopf link of unknots. The former is tight, the latter is overtwisted and the standard contact structure on is tight (a result due to Bennequin), so (+1) surgery neither preserves tightness nor necessarily creates overtwistedness.
Integral knot surgery can be interpreted in low-dimensional topology as a cobordism from to given by attaching a 4-dimensional 2-handle to M along K with framing n. This also has a geometric counterpart. A Stein cobordism between contact manifold is a cobordism X that is a complex manifold, equipped with a J-convex function f such that M, M’ are regular level sets of f.
Lemma. There is a Stein cobordism from to given by attaching a handle along each component in L with framing (-1).
Theorem. A cobordism from to is Stein if and only if it can be obtained by Legendrian surgery on a link in [edit: plus 1-handles].
For details, see Cieliebak and Eliashberg’s book.
A contact structure is Stein fillable if there exists a Stein bordism with boundary . Stein fillability is the strongest notion of fillability (see Laura’s post) and all fillable contact 3-manifolds are tight.
Proposition. Legendrian surgery preserves fillability.
So, Legendrian surgery preserves a large subclass of tight contact structures. Yet there are tight contact 3-manifolds that are not fillable. For example, (+1) surgery on the max TB right-handed trefoil in . What about them?
In addition, Legendrian surgery seems to make things more tight. Consider the following lemma:
Lemma. There is a Stein cobordism from every contact 3-manifold to a Stein fillable contact structure.
Proof. (-1) and (+1) surgeries cancel each other out, topologically and geometrically. By Ding-Geiges, there exists a link such that surgery on gives . Now performing (-1) surgery on removes all of the (+1) surgeries, leaving a contact manifold obtained by (-1) surgery on a Legendrian link in the Stein fillable , hence it is Stein fillable.
This leads us to the following conjecture:
Conjecture. The contact structure given by Legendrian surgery on a Legendrian knot in a tight contact structure is tight.