Bar-Natan Homology for null homologous links in $\mathbb{R}\mathbb{P}$^3

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