Non-homotopic Loops with a Bounded Number of Pairwise Intersections

TL;DR

This paper establishes an asymptotically tight exponential upper bound on the number of non-homotopic loops with limited intersections in the plane, improving previous bounds significantly.

Contribution

It introduces a new exponential bound for the maximum size of such loop families, extending the understanding of topological complexity in planar point sets.

Findings

Bound of e^{O(√k)} for n=2 case

Improved upon previous bound 2^{(2k)^4}

Established relation between cases with x in V_n and outside

Abstract

Let $V_n$ be a set of $n$ points in the plane and let $x \notin V_n$. An $x$-loop is a continuous closed curve not containing any point of $V_n$. We say that two $x$-loops are non-homotopic if they cannot be transformed continuously into each other without passing through a point of $V_n$. For $n=2$, we give an upper bound $e^{O\left(\sqrt{k}\right)}$ on the maximum size of a family of pairwise non-homotopic $x$-loops such that every loop has fewer than $k$ self-intersections and any two loops have fewer than $k$ intersections. The exponent $O\big(\sqrt{k}\big)$ is asymptotically tight. The previous upper bound bound $2^{(2k)^4}$ was proved by Pach, Tardos, and T\'oth [Graph Drawing 2020]. We prove the above result by proving the asymptotic upper bound $e^{O\left(\sqrt{k}\right)}$ for a similar problem when $x \in V_n$, and by proving a close relation between the two problems.