Contact 3-manifolds that admit a non-free toric action

TL;DR

This paper classifies contact toric 3-manifolds and describes contact structures on 3-manifolds that serve as concave boundaries of linear sphere plumbing, advancing understanding of contact geometry in low dimensions.

Contribution

It provides an explicit classification of contact toric 3-manifolds and characterizes contact structures on certain 3-manifolds as boundary components of linear plumbing.

Findings

Complete classification of contact toric 3-manifolds

Explicit descriptions of contact structures on boundary manifolds

Connection between contact structures and linear plumbing configurations

Abstract

We classify contact toric 3-manifolds up to contactomorphism, through explicit descriptions, building off of work by Lerman [Lerman03]. As an application, we classify all contact structures on 3-manifolds that can be realised as a concave boundary of linear plumbing over spheres. The later result is inspired by the work [MNRSTW25].