On Odd Torsion in Even Khovanov Homology

Abstract

This short note resolves the most important part of the PS braid conjecture while introducing the first known examples of knots and links with odd torsion of order 9, 27, 81, and 25 in their even Khovanov homology.

Keywords:

• knots and links
• even Khovanov homology
• torsion

2010 Mathematics Subject Classification:

• Primary 57M25 and Secondary: 57M27

Acknowledgments

The computational data in this note was obtained using JavaKh-v2 written by Scott Morrison. It is an update of Jeremy Green’s JavaKh-v1 written under the supervision of Dror Bar-Natan. The author would like to thank Adam Lowrance, Józef H. Przytycki, Radmila Sazdanović, and Alexander N. Shumakovitch for their useful comments and suggestions on this project. The author would also like to thank Dirk Schütz for his comments on an earlier version of this paper and informing him about the presence of ${\mathbb{Z}}_9$ and ${\mathbb{Z}}_{27}$ torsion in the KH of the torus knots, T(9, 10) and T(9, 11) [13]. The author would like to thank the referee for their suggestions. The author is grateful to the Columbian College of Arts and Sciences at the George Washington University for the Dean’s Dissertation Completion Fellowship.

Declaration of Interest

No potential conflict of interest was reported by the author.