Quasi right-veering braids and non-loose links
  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. 
Tetsuya Ito and Keiko Kawamuro
math.GT