Georgia Institute of Technology College of Sciences

A knot in a contact manifold is Legendrian if it is everywhere tangent to the contact planes. The classification problem in Legendrian knot theory has always generated significant interest. The problem gets a lot more complicated when we consider links. In this talk, I'll survey some of the results in this area and then discuss the classification problem for cable links of uniformly thick knot type.  If time permits, I'll also mention the classification of links in the overtwisted setting. Part of this is joint work with John Etnyre, Hyunki Min, and Tom Rodewald. 

Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

A result by Ozsvath and Szabo states that the knot Floer complex of an L-space knot is a staircase. In this talk, we will discuss a similar result for two-component L-space links: the link Floer complex of such links can be thought of as an array of staircases. We will describe an algorithm to extract this array directly from the H-function of the link. As an application, we will discuss how to use this and the link surgery formula to compute the knot Floer complex and the tau-invariant of a certain class of satellite knots. This is joint work with Ian Zemke and Hugo Zhou.

Vanishing cycles of Lefschetz fibrations give examples of Lagrangian spheres in the fiber. A natural question, first raised by Donaldson, is whether all Lagrangian spheres arise this way. We focus on this problem for positive rational surfaces, which were shown to admit a geometric structure called almost toric fibrations. I will talk about a work-in-progress showing all Lagrangian spheres here are visible in an almost toric fibration and thus are vanishing cycles of a nodal degeneration.