On the Khovanov homology of 3-braids

TL;DR

This paper proves that the Khovanov homology of 3-braid closures contains only order 2 torsion, confirming a longstanding conjecture and extending previous classifications.

Contribution

It completes the proof of the torsion conjecture for all classes of 3-braids and applies Bar-Natan's techniques to establish the result.

Findings

Khovanov homology of 3-braid closures has only 2-torsion.

Confirmed the Knight-move conjecture for 3-braids.

Extended the classification to all 3-braid classes.

Abstract

We prove the conjecture of Przytycki and Sazdanovic that the Khovanov homology of the closure of a 3-stranded braid only contains torsion of order 2. This conjecture has been known for six out of seven classes in the Murasugi-classification of 3-braids and we show it for the remaining class. Our proof also works for the other classes and relies on Bar-Natan's version of Khovanov homology for tangles as well as his delooping and cancellation techniques, and the reduced integral Bar-Natan--Lee--Turner spectral sequence. We also show that the Knight-move conjecture holds for 3-braids.