A JSJ-type decomposition theorem for symplectic fillings

Abstract

We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along \emph{mixed tori} -- these are convex tori satisfying a particular geometric condition. As an application, we show that if $(M,\hat{I}¾)$ is obtained from $(S^3,\hat{I}{¾}_{\mathrm{std}})$ via Legendrian surgery along a knot which has been stabilized both positively and negatively, then $(M,\hat{I}¾)$ has a unique exact filling.

The main statements are unchanged from the previous version, but many corrections have been made to the proof based on referee feedback. A subsection addressing the role of slope in splitting along a mixed torus has been added