New torsion patterns in Khovanov homology

TL;DR

This paper identifies new patterns of torsion elements in Khovanov homology, extending previous results and applying them to various classes of knots and links to better understand their algebraic structures.

Contribution

It introduces new torsion patterns in Khovanov homology and provides methods to distinguish and analyze these torsion elements across different link families.

Findings

Identified new torsion patterns in Khovanov homology.

Determined torsion elements in small twist knots, pretzel links, and braid closures.

Developed submodule techniques to distinguish torsion elements.

Abstract

In a previous paper by the authors, we found some patterns in link diagrams that give rise to torsion elements of order two in their Khovanov homology. In this paper we extend these results by providing new torsion patterns. Many of the torsion elements found in this way have the same homological and quantum degrees; we identify a type of submodules of the Khovanov chain complex that allows us to prove that most of these torsion elements living in the same Khovanov module are really different. We use the results of this paper together with those in the previous one to find all the torsion elements in many small twists knots. In addition, we apply them to determine torsion elements in some families of pretzel links, closures of braids with three strands and rational links.