Bott-integrability of overtwisted contact structures

TL;DR

This paper characterizes when overtwisted contact structures on closed 3-manifolds admit Bott-integrable Reeb flows, linking this property to the Euler class represented by a graph link.

Contribution

It provides a necessary and sufficient condition for overtwisted contact structures to have Bott-integrable Reeb flows based on the Euler class representation.

Findings

Overtwisted contact structures admit Bott-integrable Reeb flows iff their Euler class's Poincaré dual is a graph link.

The characterization connects contact topology with the topology of the underlying manifold.

The result offers a criterion for constructing or identifying Bott-integrable Reeb flows in contact geometry.

Abstract

We show that an overtwisted contact structure on a closed, oriented 3-manifold can be defined by a contact form having a Bott-integrable Reeb flow if and only if the Poincar\'e dual of its Euler class is represented by a graph link.