Low-dimensional contact and symplectic topology

Abstract: Given an immersed, Maslov 0, exact Lagrangian filling of a Legendrian link, applying Lagrangian surgery on a transverse double point (with "vanishing index and action") produces a new Maslov 0, exact Lagrangian filling. We show that it is not always possible to "reverse" the surgery by giving examples of Legendrian knots admitting Lagrangian fillings of genus g with p transverse double points which can not come from a Lagrangian filling of genus g-1 with p+1 double points after surgery. The proof of this involves a new obstruction to the existence of embedded, Maslov 0, exact Lagrangian cobordisms, via the theory of augmentations. Namely, if there exists such a Lagrangian cobordism from L^- to L^+, we show that the number of equivalence classes of augmentations of L^- must be smaller than that of L^+. This is a joint work with Orsola Capovilla-Searle, Maÿlis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.

Abstract: Augmentations and Lagrangian fillings are two closely-related objects associated to a Legendrian. Embedded exact Lagrangian fillings of a Legendrian can induce augmentations of the Legendrian. But not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we introduce immersed exact Lagrangian fillings into the picture and show that all the augmentations come from possibly immersed exact Lagrangian fillings. This is a joint work with Dan Rutherford.

Abstract: In the study of Legendrian links, one important task is to distinguish different exact Lagrangian fillings of a Legendrian link, up to Hamiltonian isotopy, in symplectic R^4. We introduce a cluster K2 structure on the augmentation variety of the Chekanov-Eliashberg dga for the rainbow closure of any positive braid. Using the Eckholm-Honda-Kalman functor from the cobordism category of Legendrian links to the category of dga’s, we prove that a big family of fillings give rise to cluster seeds on the augmentation variety, and these cluster seeds can be used to distinguish non-Hamiltonian isotopic fillings. Moreover, by relating a cyclic rotation concordance with the cluster Donaldson-Thomas transformation on the augmentation variety, we prove that, except for a family of positive braids that are associated with finite type quivers, the rainbow closure of all other positive braids admit infinitely many non-Hamiltonian isotopic fillings. This is joint work with H. Gao and L. Shen.

Lagrangian cobordism induces a preorder on the set of Legendrian links in any contact 3-manifold. We show that any finite collection of null-homologous Legendrian links in a tight contact 3-manifold with a common rotation number have an upper bound with respect to the pre-order. In particular, we construct an exact Lagrangian cobordism from each element of the collection to a common Legendrian link. This construction allows us to define a notion of minimal Lagrangian genus between any two null-homologous Legendrian links with common rotation number.

Talk 1: Motivated by Eliashberg's 'filling by holomorphic discs,' we introduce a technique for decomposing symplectic fillings whose boundaries admit splitting surfaces. We focus particularly on the genus 1 case, and discuss the contact manifolds whose fillings might result from this decomposition.

Talk 2: Using the JSJ decomposition for symplectic fillings developed in our first talk, we complete the classification (up to diffeomorphism) of exact symplectic fillings of lens spaces. This is joint work with Youlin Li, and our result was independently obtained by John Etnyre and Agniva Roy.

Abstract: In these talks I present joint work with Leonor Godinho and Daniele Sepe towards the classification of Hamiltonian circle actions on compact orbifolds of dimension 4, with isolated cyclic singularities.

 In the first part I will explain what orbifolds are and how circle actions are defined on them. 

 In the second part I will explain how to associate a graph to any such space, and show that two 4 dimensional orbifolds with Hamiltonian circle actions are isomorphic, if and only if their graphs are isomorphic. I will then give a list of minimal models from which all orbifolds with a Hamiltonian circle action can be constructed.

This work is inspired and based on the work of Yael Karshon, who classified Hamiltonian circle actions on compact manifolds of dimension four.

Abstract:  I will discuss recent work, joint with John Etnyre, where we completed the classification of symplectic fillings of lens spaces. In the first talk, I will go over the main techniques used, and work through an example.The second talk will use some notation mentioned in the first, and discuss applications of the result and research questions stemming from it. The first is towards the Legendrian Berge Conjecture, while the second is about constraining and constructing the existence of Stein cobordisms between tight lens spaces.

Abstract: A knot K in S^3 is called slice if K bounds a smoothly properly embedded disk in D^4. The sliceness of (topologically slice) knots has been studied well via gauge theory, Heegaard Floer theory and Khovanov homology theory.  In this talk, we develop a homotopy theoretic treatment of the Seiberg-Witten equation and give new constraints for sliceness of knots in general 4-manifolds with S^3 boundary rather than D^4. 

This is joint work with Nobuo Iida and Anubhav Mukherjee.

Abstract: Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. In this talk I'll show that no Legendrian knot which is both concordant to and from the unstabilized Legendrian unknot can be the closure of an index 3 braid except the unknot itself.

Abstract: Knot theory associated to overtwisted manifolds is less explored. There are two types of knots/links in an overtwisted manifold namely loose and non-loose. These knots are different than the knots in tight manifolds in many ways. In the first part of the talk, I'll discuss some background and history. In the second part,  I'll start with my recent results on classifying loose null-homologous links. Next, I'll talk about an invariant named support genus of knots and links and show that this invariant vanishes for loose links. I'll end the second part with some future work directions.

Abstract: In this talk, I present a 4-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Yx[0,1]. I begin by reviewing the 3-dimensional characterization of tightness, the Eliashberg Bennequin Inequality. Then I prove that our conjecture holds for contact 3-manifolds with non-vanishing Heegaard Floer contact invariant. This is all based on joint work with Matthew Hedden. 

Abstract: Similarly to how every smooth knot has a Legendrian representative (in fact, infinitely many different representatives), in this talk we will discuss why every ribbon cobordism has a Lagrangian representative. Meaning, if $C$ is a ribbon cobordism in $[0,1]\times S^3$ from the link $K_0$ to $K_1$, then there are Legendrian realizations $\Lambda_0$ and $\Lambda_1$ of $K_0$ and $K_1$, respectively, such that $C$ may be isotoped to a decomposable Lagrangian cobordism from $\Lambda_0$ to $\Lambda_1$. We will also give examples of some interesting constructions of such decomposable Lagrangian cobordisms. This is joint work with John Etnyre.