Stein-fillable open books of genus one that do not admit positive factorisations

TL;DR

This paper constructs an infinite family of genus one open book decompositions supporting Stein-fillable contact structures, demonstrating that their monodromies lack positive factorizations, thus clarifying the relationship between Stein fillings and open book decompositions.

Contribution

It provides the first known examples of genus one open books supporting Stein-fillable structures without positive monodromy factorizations.

Findings

Constructed an infinite family of genus one open books supporting Stein-fillable contact structures.

Proved that their monodromies do not admit positive factorizations.

Established that the correspondence between Stein fillings and positive factorizations is limited to planar open books.

Abstract

We construct an infinite family of genus one open book decompositions supporting Stein-fillable contact structures and show that their monodromies do not admit positive factorisations. This extends a line of counterexamples in higher genera and establishes that a correspondence between Stein fillings and positive factorisations only exists for planar open book decompositions.