Mixed tori in contact surgery diagrams

TL;DR

This paper introduces a diagrammatic method for applying the symplectic JSJ decomposition to classify exact and weak symplectic fillings of 3-dimensional contact manifolds, including lens spaces and plumbed manifolds.

Contribution

It develops a new diagrammatic framework for symplectic JSJ decomposition and provides an algorithm for classifying symplectic fillings of various 3-manifolds.

Findings

Recovered known classifications for lens spaces and torus bundles.

Provided an algorithm for classifying fillings of plumbed 3-manifolds.

Established a diagrammatic approach to symplectic JSJ decomposition.

Abstract

We develop a diagrammatic framework for applying the symplectic JSJ decomposition to exact/weak symplectic fillings of 3-dimensional contact manifolds. Namely, we apply the symplectic JSJ decomposition to a contact surgery diagram for some $(Y,\zeta)$, producing a finite collection of contact manifolds, also described diagrammatically, whose exact/weak symplectic fillings determine those of $(Y,\zeta)$. We apply this technique to recover known symplectic filling classifications for certain lens spaces and torus bundles, and also to provide an algorithm for classifying the exact/weak symplectic fillings of a large class of plumbed 3-manifolds.