  A closed braid naturally gives rise to a transverse link in the standard
contact 3-space. We study the effect of the dynamical properties of the braid
monodromy, such as right-veering, on the contact-topological properties of the
transverse link and its transverse invariants in knot Floer and Khovanov
homologies. In particular, we show that a 3-braid is right-veering if and only
if the (hat-version of) the Heegaard Floer transverse invariant is non-zero.
For higher-index braids, we show that this invariant is non-zero whenever the
braid monodromy has the fractional Dehn twist coefficient C>1. These results
clarify the qualitative meaning of the transverse invariants; our proofs use
grid diagrams and the structure of Dehornoy's braid ordering.
