Planar open books with four binding components

TL;DR

This paper constructs and classifies planar open books with four binding components on three-manifolds, linking them to contact structures, and provides counterexamples to existing conjectures in contact topology.

Contribution

It offers an explicit construction for all planar open books with four binding components and analyzes their contact structures, including fillability and invariants.

Findings

Every planar open book with four binding components arises from integral surgery on three-component pure braid closures.

Characterization of all symplectically fillable contact structures within a specific class of monodromies.

Counterexample to a conjecture by Honda, Kazez, Matic showing a right-veering diffeomorphism supports an overtwisted contact structure.

Abstract

We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and results on exceptional surgeries on hyperbolic links, we show that any contact structure of S^3 supports a planar open book with four binding components, determining the minimal number of binding components needed for planar open books supporting these contact structures. In addition, we study a class of monodromies of a planar open book with four binding components in detail. We characterize all the symplectically fillable contact structures in this class and we determine when the Ozsvath-Szabo contact invariant vanishes. As an application, we give an example of a right-veering diffeomorphism on the four-holed sphere which is not destabilizable and yet supports an overtwisted contact structure. This provides a counterexample to a conjecture of Honda, Kazez, Matic from arXiv:0609734 .