I am interested in extensions of Khovanov homology to links embedded in particular three-manifolds, Legendrian links, transverse links, and virtual links. I am also interested in the theory of universal sl(n) link homologies and combinatorial knot Floer homology.

My research primarily focuses on the Bar-Natan skein module, which is a skein module of three-manifolds. In a recent paper, I connected the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. My future research goals include construction of a Bar-Natan skein module for contact three-manifolds.

I am very interested in invariants of Legendrian and transverse knots as well as skein modules of contact three-manifolds. A research goal is to work on categorification of invariants of Legendrian and transverse links in arbitrary contact three-manifolds.

I am fascinated with graph theory and combinatorics. I appeal to techniques in these areas as often as possible in my own research. I also enjoy them as self-contained topics.

My research has connections with algebraic geometry, especially with the (n,n) Springer variety. I am currently conducting research with Prof. Julianna Tymoczko on this and other Springer varieties and looking at connections to representation theory, topology, and combinatorics.

Some information on Average Crossing Number (Warning: Algorithm is in need of editing!)

Notes on a lecture I gave about Springer varieties

The Bar-Natan skein module of the solid torus and the homology of (n,n) Springer varieties

   

Conference Presentation Slides #1

   

Conference Presentation Slides #2

LSU/UI virtual seminar presentation on Plamenevskaya’s paper: Transverse knots and

    Khovanov homology.