A spanning tree model for Khovanov homology using algebraic Morse theory and its applications

Swarup Kumar Das (TCG CREST)

Abstract: The spanning tree complex of the Tait graph associated to a knot diagram was defined by Champanerkar and Kofman but an explicit combinatorial form of the differential remained unknown. In this talk, we will describe an alternate formulation of the spanning tree complex using algebraic Morse theory and describe the differential using graph theoretic information of the Tait graph. As an application, we reprove a result due to Shumakovitch regarding the existence of 2-torsion in Khovanov homology. We will also describe how to interpret Rasmussen's s-invariant using the spanning tree model and discuss some of its applications. This talk is based on a joint work with Aninda Banerjee and Apratim Chakraborty.