The arc complex and contact geometry: non-destabilizable planar open book decompositions of the tight contact 3-sphere

TL;DR

This paper introduces the arc complex as a new tool to analyze open book decompositions in contact geometry, demonstrating the existence of non-destabilizable planar open books with more than four boundary components in the tight contact 3-sphere.

Contribution

It characterizes destabilizability via essential translation distance and constructs explicit examples of non-destabilizable planar open books with five or more boundary components.

Findings

Planar open books with 4 or fewer boundary components always destabilize.

Existence of non-destabilizable planar open books with 5 or more boundary components.

Arc complex effectively characterizes destabilizability in contact geometry.

Abstract

In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this result to show that there are planer open books of the standard contact structure on the 3-sphere with 5 (or any number larger than 5) boundary components that do not destabilize. We also show that any planar open book with 4 or fewer boundary components does destabilize.