CALDAM 2021 📺 CALDAM 2021 📺 G2OAT | tags:[ counting problems non-homotopic loops ]

# On the Intersections of Non-homotopic Loops

with Michal Opler, Matas Šileikis, and Pavel Valtr## Abstract

Let $V = {v_1, \dots, v_n}$ be a set of $n$ points in the plane and let $x \in V$.
An *$x$-loop* is a continuous closed curve not containing any point of $V$, except of passing exactly once through the point $x$.
We say that two $x$-loops are *non-homotopic* if they cannot be transformed continuously into each other without passing through a point of $V$.
For $n=2$, we give an upper bound $2^{O(k)}$ 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.
This result is inspired by a very recent result of Pach, Tardos, and Tóth who proved the upper bounds $2^{16k^4}$ for the slightly different scenario when $x\not\in V$.