Search for torsion in Khovanov homology

TL;DR

This paper demonstrates the existence of infinite families of links with various torsion types in their Khovanov homology, including some counterexamples to existing conjectures, expanding understanding of torsion phenomena in knot theory.

Contribution

It proves the existence of infinite families of links with specific torsion in Khovanov homology and introduces new counterexamples to the PS braid conjecture.

Findings

Infinite families of links with $bZ_n$-torsion for 2<n<9.

Links with $bZ_{2^s}$-torsion for s<24.

Counterexamples to the PS braid conjecture with 4-braid links.

Abstract

In the Khovanov homology of links, presence of $\mathbb{Z}_2$-torsion is a very common phenomenon. Finite number of examples of knots with $\mathbb{Z}_n$-torsion for $n>2$ were also known, none for $n>8$. In this paper, we prove that there are infinite families of links whose Khovanov homology contains $\mathbb{Z}_n$-torsion for $2 < n < 9$ and $\mathbb{Z}_{2^s}$-torsion for $s < 24$. We also introduce $4$-braid links with $\mathbb{Z}_3$-torsion which are counterexamples to the PS braid conjecture. We also provide an infinite family of knots with $\mathbb{Z}_5$-torsion in reduced Khovanov homology and $\mathbb{Z}_3$-torsion in odd Khovanov homology.