On the virtual Rasmussen invariant

TL;DR

This paper develops chain-level generators for the virtual Lee complex to derive bounds on the virtual Rasmussen invariant, demonstrating tight bounds for certain diagrams and proving its additivity under connect sum.

Contribution

It introduces chain-level generators for the virtual Lee complex and translates classical bounds to the virtual setting, also proving additivity of the invariant.

Findings

Bounds on the virtual Rasmussen invariant are tight for specific diagrams.

The virtual Rasmussen invariant is additive under connect sum.

Chain-level generators facilitate the computation of the virtual Lee complex.

Abstract

We produce chain-level generators of the virtual Lee complex $ Kh ' (V ) $ and use them to convert the computable bounds on the Rasmussen invariant of classical knots due to Kawamura and Lobb into bounds on the virtual Rasmussen invariant as defined by Dye, Kaestner, and Kauffman. We also exhibit a class of diagrams for which the bounds are tight. In addition, we use the chain-level generators to show that the virtual Rasmussen invariant is additive with respect to connect sum.