Transverse link invariants from the deformations of Khovanov $\mathfrak{sl}_{3}$-homology

TL;DR

This paper introduces a new family of transverse braid invariants derived from $ ext{sl}_3$-homology deformations, analyzing their properties and relating them to existing invariants, with applications to Bennequin inequalities.

Contribution

It defines the $eta_3$-invariants using the Mackaay-Vaz approach, extending the set of tools for studying transverse links in contact topology.

Findings

The $eta_3$-invariants include Wu's $ ext{psi}_3$-invariant.

Vanishing of $eta_3$-invariants relates to other known invariants.

Applications to Bennequin-type inequalities are demonstrated.

Abstract

In this paper we will make use of the Mackaay-Vaz approach to the universal $\mathfrak{sl}_3$-homology to define a family of cycles (called $\beta_3$-invariants) which are transverse braid invariants. This family includes Wu's $\psi_{3}$-invariant. Furthermore, we analyse the vanishing of the homology classes of the $\beta_3$-invariants and relate it to the vanishing of Plamenevskaya's and Wu's invariants. Finally, we use the $\beta_3$-invariants to produce some Bennequin-type inequalities.