Properties of contact toric structures and concave boundaries of linear plumbings

TL;DR

This paper investigates the geometric and topological properties of contact structures on linear plumbing boundaries, revealing their toric nature and analyzing algebraic torsion via embedded contact homology.

Contribution

It demonstrates that all such contact manifolds possess a global contact toric structure and develops methods to compute algebraic torsion and contact invariants.

Findings

All contact manifolds in this setting have a global contact toric structure.

Criteria for tight versus overtwisted contact structures are identified.

New toolkit for analyzing pseudoholomorphic curves and algebraic torsion in these contexts.

Abstract

We consider plumbings of symplectic disk bundles over spheres admitting concave contact boundary, with the goal of understanding the geometric properties of the boundary contact structure in terms of the data of the plumbing. We focus on the linear plumbing case in this article. We study the properties of the contact structure using two different sets of tools. First, we prove that all such contact manifolds have a global contact toric structure, and use tools from toric geometry to identify when the contact structure is tight versus overtwisted. Second, we study algebraic torsion measurements from embedded contact homology for these concavely induced contact manifolds, which has largely been unexplored. We develop a toolkit establishing existence and constraints of pseudoholomorphic curves adapted to the Morse-Bott Reeb dynamics of these plumbing examples, to provide the algebraic torsion and contact invariant calculations for the concave boundaries of linear plumbings.