Morse diagrams, Murasugi sums, and the mapping class group

TL;DR

This paper explores combinatorial Morse structures and diagrams for surfaces with boundary, their behavior under open book connect sums, and classifies Morse diagrams for one-holed torus pages, with applications to contact structures.

Contribution

It introduces a diagrammatic criterion for overtwisted contact structures and classifies Morse diagrams for a specific class of open books, advancing understanding of surface and 3-manifold topology.

Findings

Morse diagrams determine open book decompositions and contact structures.

A criterion for detecting overtwisted contact structures is developed.

Complete classification of Morse diagrams for one-holed torus pages is achieved.

Abstract

A combinatorial Morse structure encodes a mapping class for a surface with boundary, and the data may be efficiently represented via a Morse diagram. This diagram determines an open book decomposition of a 3-manifold, and hence, a contact structure on that 3-manifold. We examine how combinatorial Morse structures behave under the connect sum of open books, with particular attention paid to the case of negative stabilisation. This leads to a diagrammatic criterion for detecting overtwisted contact structures. Finally, in the case of open books with one-holed torus pages, we classify all the Morse diagrams associated to a fixed open book decomposition.