Positive flow-spines and contact 3-manifolds

A flow-spine of a 3-manifold is spine admitting flow that transverse to the spine, where in complement diffeomorphic constant an open ball. We say contact structure on closed, connected, oriented supported by if it has form whose Reeb flow-spine. It known Thurston and Winkelnkemper any book decomposition closed supports structure. In this paper, we introduce notion positivity for flow-spines prove positive uniquely up isotopy. The condition critical existence unique, structure, which also proved paper.