The genus zero, 3-component fibered links in $S^3$

TL;DR

This paper provides a purely topological classification of genus zero, 3-component fibered links in the 3-sphere, using Stallings twists and Heegaard diagrams, expanding understanding of open book decompositions.

Contribution

It offers a topological derivation of the classification of certain open book decompositions in $S^3$, identifying all such links via Stallings twists and Heegaard diagrams.

Findings

All such links can be constructed from a connected sum of two Hopf links.

The monodromies of these links are uniquely determined by the pair-of-pants open book structure.

The classification aligns with contact geometric results, now derived topologically.

Abstract

The open book decompositions of the 3-sphere whose pages are pairs of pants have been fully understood for some time, through the lens of contact geometry. The purpose of this note is to exhibit a purely topological derivation of the classification of such open books, in terms of the links that form their bindings and the corresponding monodromies. We construct all of the links and their pair-of-pants fiber surfaces from the simplest example, a connected sum of two Hopf links, through performing (generalized) Stallings twists. Then, by applying the now-classical theory of genus two Heegaard diagrams in $S^3$, we verify that the monodromies of the links in this family are the only ones corresponding to pair-of-pants open book decompositions of $S^3$.