Brieskorn spheres with two fillable contact structures

TL;DR

This paper classifies Brieskorn spheres with up to two symplectically fillable contact structures, based on recent classification results for tight contact structures on certain Seifert fibered spaces.

Contribution

It provides a complete list of Brieskorn spheres with at most two fillable contact structures, extending the understanding of contact topology on these manifolds.

Findings

Identified all Brieskorn spheres with up to two fillable structures

Connected classification of contact structures to Seifert fibered spaces

Extended previous results on tight contact structures

Abstract

Applying our recent classification of negative-twisting tight contact structures on Seifert fibered spaces whose base orbifold is a sphere, we provide the complete list of all the Brieskorn spheres carrying at most two symplectically fillable structures, up to isotopy, compatible with a given orientation.