Some link homologies in ${\mathbb{R}\mathbb{P}}^3$

We introduce extensions of Khovanov homology and the Lee and Bar-Natan spectral sequences for links in ${\mathbb{R}\mathbb{P}}^3$. These extensions are distinct to those previously defined by Asaeda-Przytycki-Sikora (and Gabrovšek’s generalization), Chen, and Manolescu-Willis. The new Lee and Bar-Natan theories each yield Rasmussen invariants (that are distinct to one another). The invariant extracted from the new Lee homology is distinct to that defined by Manolescu-Willis; it is unclear if the same is true for the new Bar-Natan homology and that defined by Chen.