Quasi-right-veering braids and nonloose links

Abstract

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 $\left(M,\xi \right)$ is nonloose if and only if every braid representative of $K$ with respect to every open book decomposition that supports $\left(M,\xi \right)$ is quasi-right-veering. We also show that several definitions of right-veering closed braids are equivalent.