My research revolves around categorification, low dimensional and applied topology, knot theory combinatorics, their interactions and applications. In knot theory I have been studying classical and quantum knot invariants as well as their categorified versions. This included developing software LinKnot to study links, knots, and their invariants in families, understanding torsion in Khovanov homology of links, and studying various link polynomials.
On the combinatorics side, analogies between alternating knots and planar graphs led me to investigate several graph homology theories that emerged in the past few years. In particular, Vladimir Baranovsky and I recently proved the old conjecture of Bendersky and Gitler, and as a special case, conjecture of Khovanov that there exists a spectral sequence from the Helme-Guizon and Rong graph homology to the Eastwood-Huggett homology theory. I also studied torsion in Helme-Guizon and Rong graph homology, using combinatorics and Hochschild homology of the underlying alegebra.
My recent work and work in progress includes several categorifications of the polynomial ring and orthogonal polynomials. The latter play fundamental role in various parts of mathematics and in many applications to other sciences. Upon categorification orthogonal polynomials become objects in some new monoidal categories and their usual properties acquire categorical liftings. These monoidal categories are described diagrammatically, providing analogues and counterparts of such classical representation-theoretical objects as the Temperley-Lieb algebra, the Brauer algebra, and the nilCoxeter algebra.

Selected Publications