Positive flow-spines and contact 3-manifolds, II

In our previous paper, it is proved that for any positive flow-spine P of a closed, oriented 3-manifold M , there exists a unique contact structure supported by P up to isotopy. In particular, this defines a map from the set of isotopy classes of positive flow-spines of M to the set of isotopy classes of contact structures on M . In this paper, we show that this map is surjective. As a corollary, we show that any flow-spine can be deformed to a positive flow-spine by applying first and second regular moves successively.