Positive flow-spines and contact 3-manifolds

A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form whose Reeb flow is a flow of the flow-spine. It is known by Thurston and Winkelnkemper that any open book decomposition of a closed oriented 3-manifold supports a contact structure. In this paper, we introduce a notion of positivity for flow-spines and prove that any positive flow-spine of a closed, connected, oriented 3-manifold supports a contact structure uniquely up to isotopy. The positivity condition is critical to the existence of the unique, supported contact structure, which is also proved in the paper.
Comments: 38 pages and 25 figures. To be published in Annali di Matematica Pura ed Applicata (1923 -). This paper covers until Section 7 of the previous version arXiv:1912.05774v3 [math.GT]. The contents of the remaining sections will be covered in a separate paper