# On transverse invariants from Khovanov-type homologies

## TL;DR

This paper introduces a new family of transverse invariants from Khovanov homology deformations, explores their properties, and applies them to prime knots to derive vanishing criteria and invariance results.

## Contribution

It defines the $eta$-invariants from Bar-Natan's deformation, relates them to existing invariants, and establishes new criteria for their effectiveness and vanishing in knot theory.

## Key findings

- $eta$-invariants are equivalent to Lipshitz, Ng, and Sarkar's $ ext{ψ}^ ext{±}$ invariants.
- Derived two non-negative integers as transverse invariants from $eta$-invariants.
- Proved vanishing criteria for the Plamenevskaya invariant for all prime knots with less than 12 crossings.

## Abstract

In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called $\beta$-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar's invariants $\psi^\pm$. From the $\beta$-invariants we extract two non-negative integers which are transverse invariants (the $c$-invariants). Finally, we give several conditions which imply the non-effectiveness of the $c$-invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant $[\psi]$, and the non-effectiveness of the vanishing of $[\psi]$, for all prime knots with less than 12 crossings.