Special moves for open book decompositions of 3-manifolds

TL;DR

This paper introduces a minimal set of two moves, including a Hopf plumbing and a special twist, to relate any two open book decompositions of a 3-manifold, using tools from 4D Lefschetz fibrations and contact topology.

Contribution

It identifies a complete and minimal move set for relating open book decompositions, enhancing understanding of their equivalence via Lefschetz fibrations and contact topology.

Findings

Two moves suffice to relate any two open book decompositions.

The moves include Hopf plumbing and a special local twist.

The approach uses Lefschetz fibrations and Giroux-Goodman theorem.

Abstract

We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3-manifold. One of the moves is the usual plumbing with a positive or negative Hopf band, while the other one is a special local version of Harer's twisting, which is presented in two different (but stably equivalent) forms. Our approach relies on 4-dimensional Lefschetz fibrations, and on 3-dimensional contact topology, via the Giroux-Goodman stable equivalence theorem for open book decompositions representing homologous contact structures.