Arcs intersecting at most once

TL;DR

This paper establishes precise bounds on the maximum number of essential simple arcs and curves on punctured surfaces that intersect at most once, providing new cubic and polynomial estimates depending on the surface's Euler characteristic.

Contribution

It proves exact maximal cardinalities for sets of arcs and curves with limited intersections on punctured surfaces, including a new cubic bound and polynomial estimates.

Findings

Maximal set size of arcs on punctured surfaces is 2|chi|(|chi|+1).

Polynomial bounds for sets intersecting at most once.

Maximal arcs on punctured sphere is 1/2|chi|(|chi|+1).

Abstract

We prove that on a punctured oriented surface with Euler characteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a closed surface. We also give polynomial estimates in |chi| for sets of arcs and curves pairwise intersecting a uniformly bounded number of times. Finally, we prove that on a punctured sphere the maximal cardinality of a set of arcs starting and ending at specified punctures and pairwise intersecting at most once is 1/2|chi|(|chi|+1).