Contact surgery on the Hopf link: classification of fillings

TL;DR

This paper classifies minimal symplectic fillings of lens spaces obtained via contact surgery on the Hopf link, extending previous classifications to a broader set of tight contact structures.

Contribution

It extends the classification of symplectic fillings to new tight contact structures on lens spaces arising from contact surgery on the Hopf link.

Findings

Classified minimal symplectic fillings up to homeomorphism.

Extended Lisca's results to more tight contact structures.

Connected to previous work on fillings of L(p,1).

Abstract

Let $H\subseteq S^3$ be the two-component Hopf link. After choosing a Legendrian representative of $H$ with respect to the standard tight contact structure on $S^3$ we perform contact $(-1)$-surgery on the link itself. The manifold we get is a lens space together with a tight contact strucure on it, which depends on the chosen Legendrian representative. We classify its minimal symplectic fillings up to homeomorphism (and often up to diffeomoprhism), extending the results of Lisca which covers the case of universally tight structures and the article of Plamenevskaya - Van Horn Morris which describes the fillings of $L(p,1)$.