Stability and triviality of the transverse invariant from Khovanov homology

TL;DR

This paper investigates the properties of the transverse invariant from Khovanov homology for braids and links, establishing conditions for non-vanishing, stability under twists, and applications to quasipositivity and knot classification.

Contribution

It demonstrates the non-vanishing of the invariant for braids with high fractional Dehn twist coefficients, proves its stability under full twists, and constructs examples of knots with vanishing invariants, revealing new insights into quasipositivity.

Findings

Invariant does not vanish if fractional Dehn twist coefficient > 1

Invariant is stable under adding full twists on up to n strands

Constructs pretzel knots with vanishing invariant for all transverse representatives

Abstract

We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by Plamenevskaya for their closures, which are naturally transverse links in the standard contact $3$-sphere. For any $3$-braid $\beta$, we show that the transverse invariant of its closure does not vanish whenever the fractional Dehn twist coefficient of $\beta$ is strictly greater than one. We show that Plamenevskaya's transverse invariant is stable under adding full twists on $n$ or fewer strands to any $n$-braid, and use this to detect families of braids that are not quasipositive. Motivated by the question of understanding the relationship between the smooth isotopy class of a knot and its transverse isotopy class, we also exhibit an infinite family of pretzel knots for which the transverse invariant vanishes for every transverse representative, and conclude that these knots are not quasipositive.