Spiraling and Folding: The Topological View

TL;DR

This paper constructs specific planar and torus curves with many intersections that do not form spirals, providing a counterexample to a previous proof about string graphs.

Contribution

It introduces a novel construction of intersecting curves on the torus and plane that challenge existing assumptions about spiral formations in string graphs.

Findings

Constructed pairs of curves with at least n intersections without forming spirals

Provided counterexamples to a proof by Pach and Tóth

Demonstrated complex curve configurations on the torus and plane

Abstract

For every $n$, we construct two curves in the plane that intersect at least $n$ times and do not form spirals. The construction is in three stages: we first exhibit closed curves on the torus that do not form double spirals, then arcs on the torus that do not form spirals, and finally pairs of planar arcs that do not form spirals. These curves provide a counterexample to a proof of Pach and T\'{o}th concerning string graphs.