Properties of contact toric structures and concave boundaries of linear plumbings

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 (ECH) 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 ECH algebraic torsion and contact invariant calculations for the concave boundaries of linear plumbings.