Braid monodromy, orderings and transverse invariants

Abstract

A closed braid $\beta$ naturally gives rise to a transverse link $K$ in the standard contact $3$–space. We study the effect of the dynamical properties of the monodromy of $\beta$, such as right-veering, on the contact-topological properties of $K$ and the values of transverse invariants in Heegaard Floer and Khovanov homologies. Using grid diagrams and the structure of Dehornoy’s braid ordering, we show that $\widehat{\theta }\left(K\right)\in \widehat{HFK}\left(m\left(K\right)\right)$ is nonzero whenever $\beta$ has fractional Dehn twist coefficient $C>1$. (For a $3$–braid, we get a sharp result: $\widehat{\theta }\ne 0$ if and only if the braid is right-veering.)