Bar-Natan homology for null homologous links in $\mathbb{RP}^{3}$

Daren Chen

Vol. 334 (2025), No. 2, 233–268

DOI: 10.2140/pjm.2025.334.233

Abstract

We introduce Bar-Natan homology for null homologous links in ${ℝ\mathbb{P}}^3$ over the field of two elements. It is a deformation of the Khovanov homology in ${ℝ\mathbb{P}}^3$ defined by Asaeda, Przytycki and Sikora. We also define an $s$-invariant from this deformation using the same recipe as for links in $S^3$, and prove some genus bound using it. The key ingredient is the notion of “twisted orientation” for null homologous links and cobordisms in ${ℝ\mathbb{P}}^3$.

Keywords

Bar-Natan homology, $\mathbb{RP}^{3}$, genus bound

Mathematical Subject Classification

Primary: 57K18

Milestones

Received: 22 September 2023

Revised: 20 December 2024

Accepted: 19 January 2025

Published: 8 February 2025