Grid diagrams and Khovanov homology

TL;DR

This paper explores the computation of the Jones polynomial from grid diagrams, connects different homological definitions, and provides evidence supporting the Seidel-Smith conjecture in link invariants.

Contribution

It establishes a link between grid diagram computations, Bigelow's homological Jones polynomial, and Kauffman's definition, and proves a grading coincidence in Khovanov homology.

Findings

Jones polynomial computed from grid diagrams

Connection between Bigelow's and Kauffman's definitions

Evidence supporting the Seidel-Smith conjecture

Abstract

We explain how to compute the Jones polynomial of a link from one of its grid diagrams and we observe a connection between Bigelow's homological definition of the Jones polynomial and Kauffman's definition of the Jones polynomial. Consequently, we prove that the Maslov grading on the Seidel-Smith symplectic link invariant coincides with the difference between the homological grading on Khovanov homology and the Jones grading on Khovanov homology. We give some evidence for the truth of the Seidel-Smith conjecture.