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

TL;DR

This paper extends Bar-Natan homology to null homologous links in real projective 3-space, introducing a new s-invariant and genus bounds, building on Khovanov homology deformations.

Contribution

It introduces Bar-Natan homology for null homologous links in old{RP}^3 and defines an s-invariant with genus bounds, expanding link invariants in non-orientable 3-manifolds.

Findings

Defined Bar-Natan homology for null homologous links in old{RP}^3

Constructed an s-invariant for these links

Proved genus bounds using the s-invariant

Abstract

In this paper, we introduce Bar-Natan homology for null homologous links in \mathbb{RP}^3 over the field of two elements. It is a deformation of the Khovanov homology in \mathbb{RP}^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 \mathbb{RP}^3.