  We introduce a notion of "quasi-right-veering" for closed braids, which plays
an analogous role to "right-veering" for open books. We show that a transverse
link $K$ in a contact 3-manifold $(M,\xi)$ is non-loose if and only if every
braid representative of $K$ with respect to every open book decomposition that
supports $(M,\xi)$ is quasi-right-veering. We also show that several
definitions of "right-veering" closed braids are equivalent.
