Some link homologies in $ \mathbb{RP}^3 $

TL;DR

This paper extends Khovanov homology and spectral sequences to links in real projective 3-space, introducing new invariants that differ from existing ones and providing novel tools for link classification.

Contribution

It develops new versions of Lee and Bar-Natan homologies for links in $\mathbb{RP}^3$, distinct from prior definitions, and introduces Rasmussen invariants from these theories.

Findings

New Lee and Bar-Natan theories yield distinct Rasmussen invariants.

The Rasmussen invariant from the new Lee homology differs from that of Manolescu-Willis.

It remains unclear if the new Bar-Natan invariant differs from Chen's.

Abstract

We introduce extensions of Khovanov homology and the Lee and Bar-Natan spectral sequences for links in $ \mathbb{RP}^3 $. These extensions are distinct to those previously defined by Asaeda-Przytycki-Sikora (and Gabrov\v{s}ek'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.