Planar open book decompositions and contact structures

TL;DR

This paper discusses the relationship between planar open book decompositions and contact structures on 3-manifolds, highlighting that not all contact structures are supported by such decompositions and exploring implications for contact invariants and the Weinstein conjecture.

Contribution

It demonstrates that only overtwisted contact structures are supported by planar open books, revealing limitations and implications for contact invariants and the Weinstein conjecture.

Findings

All overtwisted contact structures are supported by planar open book decompositions.

Not all contact structures are supported by planar open books.

Implications for contact invariants and the Weinstein conjecture.

Abstract

In this note we observe that while all overtwisted contact structures on compact 3--manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to invariants of contact structures and also to the Weinstein conjecture via work of Abbas Cieliebak and Hofer.