P-bigon right-veeringness and overtwisted contact structures

A contact structure is a maximally non-integrable hyperplane field $\xi$ on an odd-dimensional manifold $M$. In $3$-dimensional contact geometry, there is a fundamental dichotomy, where a contact structure is either tight or overtwisted. Making use of Giroux's correspondence between contact structures and open books for $3$-dimensional manifolds, Honda, Kazez, and Mat\'{i}c proved that verifying whether a mapping class is right-veering or not gives a way of detecting tightness of the compatible contact structure. As a counter-part to right-veering mapping classes, right-veering closed braids have been studied by Baldwin and others. Ito and Kawamuro have shown how various results on open books can be translated to results on closed braids; introducing the notion of quasi right-veering closed braids to provide a sufficient condition which guarantees tightness.

We use the related concept of $P$-bigon right-veeringness for closed braids to show that given a $3$-dimensional contact manifold $(M, \xi)$ supported by an open book $(S, \phi)$, if $L \subset (M, \xi)$ is a non-$P$-bigon right-veering transverse link in pure braid position with respect to $(S, \phi)$, performing $0$-surgery along $L$ produces an overtwisted contact manifold $(M', \xi')$. Furthermore, if we suppose $L \subset (M, \xi)$ is a pure and non-quasi right-veering braid with respect to $(S, \phi)$, performing $p$-surgery along $L$, for $p \geq 0$, gives rise to an open book $(S', \phi')$ which supports an overtwisted contact manifold $(M', \xi')$.