Fillable structures on negative-definite Seifert fibred spaces

Abstract

We classify fillable contact structures on all negative-definite star-shaped plumbings. We show that such Seifert fibred spaces admit a unique negative maximal twisting number and compute it explicitly using the Alexander filtration in lattice cohomology, providing its first Floer-theoretic interpretation. In addition, we show that all the negative-twisting tight structures on these manifolds are induced by the Stein structures on the minimal resolution of the underlying complex surface singularity. As an application, we provide a necessary condition for a negative-definite Seifert fibred space to admit a separating contact-type embedding in a strong symplectic filling of a generalised $L$-space.