Open book decompositions of fibre sums in contact topology

TL;DR

This paper develops methods to construct open book decompositions supporting various contact structures, including tight structures on the 3-torus and those resulting from Lutz twists, advancing understanding in contact topology.

Contribution

It introduces compatible open books for fiber sums along bindings and multi-sections, providing new tools for constructing contact structures.

Findings

Supported all tight contact structures on T^3

Constructed open books for Lutz twists

Reproduced known results with new methods

Abstract

In the present paper we describe compatible open books for the fibre connected sum along binding components of open books, as well as for the fibre connected sum along multi-sections of open books. As an application the first description provides simple ways of constructing open books supporting all tight contact structures on $T^3$, recovering a result by van Horn-Morris, as well as an open book supporting the result of a Lutz twist along a binding component of an open book, recovering a result by Ozbagci--Pamuk.