Georgia Institute of Technology College of Sciences

The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston. After introducing the relevant definitions, I'll explain a bit about this connection and how I used the geometric structure on this moduli space to exactly identify the kernel of the Burau representation after evaluating its formal parameter at complex roots of unity. There will be many pictures!

The talk is based on my joint works with Maxim Prasolov and Vladimir Shastin, where we studied the relation between rectangular diagrams of links and Legendrian links. This relation allows for a complete classification of exchange classes of rectangular diagrams in terms of equivalence classes of Legendrian links and their symmetry groups. Since all rectangular diagrams of given complexity can be searched, this yields a method to algorithmically compare Legendrian links. Of course, the general algorithm has too high complexity for a practical implementation, but in some situations, the most time consuming parts can be bypassed, which allows us to confirm the non-equivalence of Legendrian knots in several previously unresolved cases.

I will survey recent progress toward Khovanov homology for links in general 3-manifolds based on categorification of $q$-series invariants labeled by Spin$^c$ structures. Much of the talk will focus on the $q$-series invariants themselves. In particular, I hope to explain how to compute them for a general 3-manifold and to describe some of their properties, e.g. relation to other invariants labeled by Spin or Spin$^c$ structures, such as Turaev torsion, Rokhlin invariants, and the "correction terms'' of the Heegaard Floer theory. There are many problems to work on in this relatively new research area. If time permits, I will outline some of them, and, in the context of plumbed 3-manifolds, comment on the relation to lattice cohomology proposed by Akhmechet, Johnson, and Krushkal.

I will discuss recent work with K. Honda and Y. Huang on proving the Giroux correspondence between contact structures and open book decompositions. Though our work extends to all dimensions (with appropriate adjectives), this talk will focus on the 3-dimensional proof. I will first recall Giroux’s argument for existence of supporting open book decompositions, formulating it in the language adapted to our proof. The rest of the talk will be spent describing the proof of the stabilization correspondence.