# Torsion in thin regions of Khovanov homology

## TL;DR

This paper proves that under certain conditions, the Khovanov homology of links supported on two diagonals contains only  extsubscript{2} torsion, and verifies this for an infinite family of 3-braids, partially confirming a conjecture.

## Contribution

It establishes a local criterion for  extsubscript{2} torsion in Khovanov homology and applies it to a broad class of 3-braids, extending previous results.

## Key findings

- Links supported on two diagonals have only  extsubscript{2} torsion under mild conditions.
- An infinite family of 3-braids, including all 3-strand torus links, satisfies these conditions.
- Explicit computations of integral Khovanov homology are provided for this family.

## Abstract

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported in two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb{Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of 3-braids, strictly containing all 3-strand torus links, thus giving a partial answer to Sazdanovic and Przytycki's conjecture that 3-braids have only $\mathbb{Z}_2$ torsion in Khovanov homology. We also give explicit computations of integral Khovanov homology for all links in this family.