On the Intersections of Non-homotopic Loops

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 $x\not \in V$.

This research was initiated during the workshop KAMAK 2020 in Kytlice in Sept. 20–25, 2020. MŠ was supported by the Czech Science Foundation, grant number GJ20-27757Y, with institutional support RVO: 67985807. VB acknowledges the support of the OP VVV MEYS funded project CZ.02.1.01/0.0/0.0/16_019/0000765 “Research Center for Informatics”. PV and MO were supported by project 18-19158S of the Czech Science Foundation.