Reeb periodic orbits after a bypass attachment

TL;DR

This paper models how bypass attachments on 3D contact manifolds create Reeb periodic orbits, computes their contact homology, and provides explicit descriptions of these dynamical features.

Contribution

It introduces a model for analyzing Reeb orbits after bypass attachments and computes contact homology in these modified structures.

Findings

Reeb orbits are described via Reeb chords of the attachment arc.

Contact homology is explicitly computed for neighborhoods after bypass.

The results apply to contact structures on solid tori.

Abstract

On a 3-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.