Surgery diagrams for horizontal contact structures (Q949879)

The author considers horizontal contact structures on non-positive plumbing trees of oriented circle bundles over spheres with negative Euler numbers. Here, a contact structure on such a manifold is said to be horizontal if the contact planes (oriented by the differential of the contact form) are positively transverse to the fibers of the bundles involved in the plumbing, and the plumbing tree is said to be non-positive if for every vertex of the tree the sum of the degree of the vertex and the Euler number of the bundle corresponding to that vertex is non-positive. The author's aim in this paper is to give a new description of such contact stuctures up to isomorphism, by describing Legendrian surgery diagrams for these structures. The author gives an application to the study of Milnor fillable contact 3-manifolds, that is, contact 3-manifolds isomorphic to the contact boundary of an isolated complex surface singularity. By a result of \textit{C. Caubel, A. Némethi} and \textit{P. Popescu-Pampu} [Topology 45, No.~3, 637--689 (2006; Zbl 1098.53064)] any 3-manifold admits, up to isomorphism, at most one Milnor fillable contact structure. As an application of his constructions, the author obtains a description of the Milnor fillable contact structures of some Milnor fillable 3-manifolds.