Articles (Published/Preprint)

1. A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the 4-ball

   with Aninda Banerjee and Apratim Chakraborty

   Preprint (2025) -- (arxiv)

   The checkerboard coloring of knot diagrams offers a graph-theoretical approach to address topological questions. Champanerkar and Kofman defined a complex generated by the spanning trees of a graph obtained from the checkerboard coloring whose homology is the reduced Khovanov homology. Notably, the differential in their chain complex was not explicitly defined. We explicitly define the combinatorial form of the differential within the spanning tree complex. We additionally provide a description of Rasmussen's $s-$invariant within the context of the spanning tree complex. Applying our techniques, we identify a new infinite family of knots where each of them bounds a set of exotic discs within the $4-$ball.

2. A spanning tree model for chromatic homology

   with Aninda Banerjee, Apratim Chakraborty, and Pravakar Paul

   Preprint (2025) -- (arxiv)

   After the discovery of Khovanov homology, which categorifies the Jones polynomial, an analogous categorification of the chromatic polynomial, known as chromatic homology, was introduced. Its graded Euler characteristic recovers the chromatic polynomial. In this paper, we present a spanning tree model for the chromatic complex, i.e., we describe a chain complex generated by certain spanning trees of the graph that is chain homotopy equivalent to the chromatic complex. We employ the spanning tree model over $\mathcal{A}_m = \mathbb{Z}[X]/\langle X^m \rangle$ algebra to answer two open questions. First, we establish the conjecture posed by Sazdanovic and Scofield regarding the homological span of chromatic homology over $\mathcal{A}_m$ algebra, demonstrating that for any graph $G$ with $v$ vertices and $b$ blocks, the homological span is $v-b$. Additionally, we prove a conjecture of Helme-Guizon, Przytycki, and Rong concerning the existence of torsion of order dividing $m$ in chromatic homology over $\mathcal{A}_m$ algebra.