# On the complexity of non-orientable Seifert fibre spaces

## TL;DR

This paper provides a combinatorial framework for non-orientable Seifert fibre spaces and establishes an upper bound for their complexity based on their invariants, extending previous results to the non-orientable case.

## Contribution

It introduces a comprehensive combinatorial description for all Seifert fibre spaces, including non-orientable cases, and derives a sharp upper bound for their complexity.

## Key findings

- Provided a combinatorial description for non-orientable Seifert fibre spaces.
- Extended complexity bounds to non-orientable cases.
- Unified description covering orientable and non-orientable, closed and with boundary cases.

## Abstract

In this paper we deal with Seifert fibre 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.