Morse matchings and Khovanov homology of 4-strand torus links

TL;DR

This paper introduces a new Morse theoretic approach to simplify Khovanov homology calculations for 4-strand torus links, revealing new torsion phenomena and supporting existing conjectures.

Contribution

It presents a novel Morse theoretic method for Khovanov homology, providing computational insights into 4-strand torus links and their torsion properties.

Findings

Computed non-trivial Khovanov homology groups for all degrees

Discovered abundant 4-torsion in the homology groups

Confirmed conjecture of Gorsky, Oblomkov, and Rasmussen at the limit

Abstract

Given a link or a tangle diagram, we define algorithmic Morse theoretic simplifications on their Khovanov homology. In contrast to Bar-Natan's scanning algorithm, the cancellations are postponed until the end and performed in one go. Although our novel approach is computationally inferior to Bar-Natan's algorithm, it side-steps the need for a large amount of iterations, making it more fitting for theoretical analysis. Our main application is towards integral Khovanov homology of 4-strand torus links, for which we compute non-trivial Khovanov homology groups in all homological degrees and find an abundance of $4$-torsion. At the limit $T(4,\infty)$, our computations agree with a conjecture of Gorsky, Oblomkov and Rasmussen. For finite $n$, we use the $\lambda$-invariant of Lewark, Marino and Zibrowius to derive lower bounds on proper rational Gordian distances from $T(4,n)$.