Non-homotopic Loops with a Bounded Number of Pairwise Intersections
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Let V_n be a set of n points in the plane and let x ∉ 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(√(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. The exponent O(√(k)) is asymptotically tight. The previous upper bound bound 2^(2k)^4 was proved by Pach, Tardos, and Tóth [Graph Drawing 2020]. We prove the above result by proving the asymptotic upper bound e^O(√(k)) for a similar problem when x ∈ V_n, and by proving a close relation between the two problems.
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