On the Classification of Planar Contact Structures

TL;DR

This paper investigates contact structures supported by planar open book decompositions, analyzing overtwistedness via right-veering diffeomorphisms, and provides examples and fillability results for specific cases.

Contribution

It introduces methods to track overtwistedness in planar contact structures and constructs explicit examples supported by four-punctured sphere open books.

Findings

Infinitely many overtwisted contact structures supported by four-punctured sphere open books.

Identification of a family of contact structures that are holomorphically fillable.

Application of lantern relation to prove fillability.

Abstract

In this paper, we focus on contact structures supported by planar open book decompositions. We study right-veering diffeomorphisms to keep track of overtwistedness property of contact structures under some monodromy changes. As an application we give infinitely many examples of overtwisted contact structures supported by open books whose pages are the four-punctured sphere, and also we prove that a certain family is holomorphically fillable using lantern relation.