2013/02/26 by Jiřı́ Matoušek, Matoušek, Jiří, Eric Sedgwick +5 · 1 citation
Computer Science · Mathematics · #52C45 #57N05 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1302.6475
openalex publication_date 2013/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider two systems of curves (α1,...,αm) and (β1,...,βn) drawn on a compact two-dimensional surface M with boundary. Each αi and each βj is either an arc meeting the boundary of M at its two endpoints, or a closed curve. The αi are pairwise disjoint except for possibly sharing endpoints, and similarly for the βj. We want to "untangle" the βj from the αi by a self-homeomorphism of M; more precisely, we seek a homeomorphism ϕ:M→ M fixing the boundary of M pointwise such that the total number of crossings of the αi with the ϕ(βj) is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3-manifolds. We prove that if M is planar, i.e., a sphere with h≥ 0 boundary components ("holes"), then O(mn) crossings can be achieved (independently of h), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface M with h holes and of (orientable or nonorientable) genus g, we obtain an O((m+n)4) upper bound, again independent of h and g. The proofs rely, among others, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov.