Stability and triviality of the transverse invariant from Khovanov homology

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.

arXiv comment23 pages, 8 figures. Updated to most recent version

References in corpus (18)

• Knot Floer homology and the four-ball genus
• Holomorphic disks and genus bounds
• On Khovanov's categorification of the Jones polynomial
• On the Khovanov and knot Floer homologies of quasi-alternating links
• Legendrian knots, transverse knots and combinatorial Floer homology
• Markov theorem for transversal links
• Heegaard Floer homology and genus one, one boundary component open books
• Homological thickness and stability of torus knots
• The Markov Theorem for transverse knots
• On arc index and maximal Thurston-Bennequin number
• A Legendrian Thurston-Bennequin bound from Khovanov homology
• Knots with identical Khovanov homology
• Branched covers of quasipositive links and L-spaces
• On transverse invariants from Khovanov homology
• On the Mahler measure of Jones polynomials under twisting
• The rational Khovanov homology of 3-strand pretzel links
• An annular refinement of the transverse element in Khovanov homology
• On transverse invariants from Khovanov-type homologies

Cited by in corpus (1)

• Applications of higher-dimensional Heegaard Floer homology to contact topology