On the complexity of non-orientable Seifert fibre spaces

In this paper we deal with Seifert bre spaces, which are compact 3-manifolds admitting a foliation by circles. We give a combinatorial description for these manifolds in all the possible cases: orientable, non-orientable, closed, with boundary. Moreover, we compute a potentially sharp upper bound for their complexity in terms of the invariants of the combinatorial description, extending to the non-orientable case results by Fominykh and Wiest for the orientable case with boundary and by Martelli and Petronio for the closed orientable case. Our upper bound is indeed sharp for all Seifert bre spaces contained in the census of non-orientable closed 3-manifolds classied with respect to complexity.