Hyperbolic Knots and Torsion in Khovanov Homology

TL;DR

This paper demonstrates that the presence of torsion in the Khovanov homology of a knot implies infinitely many hyperbolic and satellite knots share this property, providing new examples of hyperbolic knots with non-$bZ_2$ torsion.

Contribution

It establishes a link between torsion in Khovanov homology and the existence of infinitely many hyperbolic and satellite knots with similar torsion properties, including the first known examples with odd torsion.

Findings

Infinite families of hyperbolic knots with $bZ_m$ torsion in Khovanov homology.

Existence of prime satellite knots with $bZ_m$ torsion.

First known examples of hyperbolic knots with non-$bZ_2$ torsion.

Abstract

In this note, we show that if there is a knot in $S^3$ having $\mathbb{Z}_m$ torsion in its Khovanov homology, then there are infinitely many hyperbolic knots and infinitely many prime satellite knots having $\mathbb{Z}_m$ torsion in their Khovanov homology. As an application, we give the first known examples of hyperbolic knots and links with odd (and other non-$\mathbb{Z}_2$ torsion) in their Khovanov homology.