A Rasmussen invariant for links in $\mathbb{RP}^3$

TL;DR

This paper extends Khovanov homology to links in real projective 3-space, constructing a Rasmussen-type invariant that constrains link cobordisms and distinguishes certain knot concordance properties.

Contribution

It introduces a Lee-type deformation of Khovanov homology for $ ext{RP}^3$ and defines a new Rasmussen s-invariant in this setting, providing new tools for knot theory in non-orientable 3-manifolds.

Findings

The s-invariant constrains the genera of link cobordisms in $ ext{RP}^3$.

Examples of knots in $S^3$ are given that are concordant but not equivariantly concordant.

The theory extends Rasmussen's invariant to a new topological setting.

Abstract

Asaeda-Przytycki-Sikora, Manturov, and Gabrov\v{s}ek extended Khovanov homology to links in $\mathbb{RP}^3$. We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen's s-invariant in this setting. We show that the s-invariant gives constraints on the genera of link cobordisms in the cylinder $I \times \mathbb{RP}^3$. As an application, we give examples of freely 2-periodic knots in $S^3$ that are concordant but not standardly equivariantly concordant.