Large 1-systems of Curves in Non-orientable Surfaces

TL;DR

This paper extends the study of collections of curves with limited intersections from orientable to non-orientable surfaces, providing new bounds and constructions in the latter setting.

Contribution

It generalizes a known construction to non-orientable surfaces, establishing lower bounds for large collections of curves with restricted intersections.

Findings

Established a lower bound for the maximum number of such curves in non-orientable surfaces.

Extended the Malestein-Rivin-Theran construction to non-orientable cases.

Provided new insights into curve arrangements in non-orientable topology.

Abstract

A longstanding avenue of research in orientable surface topology is to create and enumerate collections of curves in surfaces with certain intersection properties. We look for similar collections of curves in non-orientable surfaces. A surface is non-orientable if and only if it contains a M\"obius band. We generalize a construction of Malestein-Rivin-Theran to non-orientable surfaces to exhibit a lower bound for the maximum number of curves that pairwise intersect 0 or 1 times in a generic non-orientable surface.