Systems of curves on non-orientable surfaces

TL;DR

This paper investigates the maximum size of certain loop systems on non-orientable surfaces, establishing asymptotic and exact results, and introduces new bounds for intersecting arc systems.

Contribution

It provides the first asymptotic estimate for maximal complete 1-systems of loops on non-orientable surfaces and determines exact counts for punctured projective planes.

Findings

Maximal complete 1-systems of loops have size proportional to ||^2.

Exact cardinality for punctured projective planes.

Maximal systems of arcs intersecting at most once have size 2||(||+1).

Abstract

We show that the order of the cardinality of maximal complete $1$-systems of loops on non-orientable surfaces is $\sim |\chi|^{2}$. In particular, we determine the exact cardinality of maximal complete $1$-systems of loops on punctured projective planes. To prove these results, we show that the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is $2|\chi|(|\chi|+1)$.