Unknotting number and Khovanov homology

Akram Alishahi

Vol. 301 (2019), No. 1, 15–29

DOI: 10.2140/pjm.2019.301.15

Abstract

We show that the order of $h$-torsion homology classes in the Bar-Natan deformation of Khovanov homology with $\mathbb{Z}∕2\mathbb{Z}$-coefficients is a lower bound for the unknotting number. This is not a bound for the slice genus, unlike most lower bounds for the unknotting number, and only vanishes for the unknot. We give examples of knots for which this is a better lower bound than $|s\left(K\right)∕2|$, where $s\left(K\right)$ is the Rasmussen $s$ invariant defined by the Bar-Natan spectral sequence.

Keywords

unknotting number, Khovanov homology, Bar-Natan homology

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

Milestones

Received: 3 June 2018

Revised: 14 September 2018

Accepted: 15 November 2018

Published: 16 September 2019