By a famous Giroux correspondence, there is an one-to-one correspondence between the set of contact 3-manifolds and the set of open books (modulo stabilizations). Therefore it is a fundamental problem to translate a property of contact 3-manifolds into a property of open books (or vice versa).
In this talk we focus on the right-veering property and its variant. Honda-Kazez-Matic showed that the right-veering property of open books characterizes tightness: A contact 3-manifold is tight if and only if every open book decomposition is right-veering. A similar result is not true for closed braids, however: Right-veering closed braids do not characterize non-loose-ness, (a counterpart of tightness property).
In this talk we discuss contact geometrical property related to right-veering closed braids. Among them, we show that not-right-veering closed braids are virtually loose, i.e. some finite coverings of its complement are overtwisted)
This is a joint work with Keiko Kawamuro (Univ. Iowa)