Vol. 301, No. 1, 2019

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Unknotting number and Khovanov homology

Akram Alishahi

Vol. 301 (2019), No. 1, 15–29
Abstract

We show that the order of $h$-torsion homology classes in the Bar-Natan deformation of Khovanov homology with $ℤ∕2ℤ$-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