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

TL;DR

This paper introduces twist left-veering mapping classes for punctured surfaces, demonstrating their connection to overtwisted contact structures and characterizing looseness of associated closed braids.

Contribution

It defines twist left-veering mapping classes and establishes their role in supporting overtwisted contact structures and in determining the looseness of closed braids.

Findings

Twist left-veering open books support overtwisted contact structures.

Criteria for when the closed braid from punctures is loose or virtually loose.

Characterization of twist left-veering classes in relation to contact topology.

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.