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.
- LinKnot- Knot Theory by Computer, World Scientific, joint with S. Jablan
- Graph homology and graph configuration spaces with V. Baranovsky
- Categorification of the polynomial ring with M. Khovanov
- On the first group of the chromatic cohomology of graphs, Geometriae Dedicata,Vol 140, No. 1 (2009)19-48, joint with M. Pabiniak and J. Przytycki.
- Braid Family Representatives, Journal of Knot Theory and Its Ramifications 17 (7) (2008) 817-833, joint with S. Jablan
- Unlinking number and unlinking gap, Journal of Knot Theory and Its Ramifications 16 (10) (2007) 1331-1355, joint with S. Jablan