Twist left-veering open books, overtwistedness, looseness and virtual looseness

Abstract

We introduce twist left-veering mapping classes of punctured surfaces. We prove that a twist left-veering open book supports an overtwisted contact structure and determine when the closed braid coming from the punctures is loose or virtually loose.

25 pages, 21 Figures; Title is changed, results on virtual looseness and depth of closed braids are added