Untangling two systems of noncrossing curves

Matoušek J, Sedgwick E, Tancer M, Wagner U. 2016. Untangling two systems of noncrossing curves. Israel Journal of Mathematics. 212(1), 37–79.


Journal Article | Published | English

Scopus indexed
Author
Matoušek, Jiří; Sedgwick, Eric; Tancer, MartinIST Austria ; Wagner, UliIST Austria
Department
Abstract
We consider two systems (α1, …, αm) and (β1, …,βn) of simple curves 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 ai 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 ai 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 ≥ 0, we obtain an O((m + n)4) upper bound, again independent of h and g. The proofs rely, among other things, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov.
Publishing Year
Date Published
2016-05-01
Journal Title
Israel Journal of Mathematics
Acknowledgement
Supported by the ERC Adv anced Grant No. 267165.
Volume
212
Issue
1
Page
37 - 79
IST-REx-ID

Cite this

Matoušek J, Sedgwick E, Tancer M, Wagner U. Untangling two systems of noncrossing curves. Israel Journal of Mathematics. 2016;212(1):37-79. doi:10.1007/s11856-016-1294-9
Matoušek, J., Sedgwick, E., Tancer, M., & Wagner, U. (2016). Untangling two systems of noncrossing curves. Israel Journal of Mathematics. Springer. https://doi.org/10.1007/s11856-016-1294-9
Matoušek, Jiří, Eric Sedgwick, Martin Tancer, and Uli Wagner. “Untangling Two Systems of Noncrossing Curves.” Israel Journal of Mathematics. Springer, 2016. https://doi.org/10.1007/s11856-016-1294-9.
J. Matoušek, E. Sedgwick, M. Tancer, and U. Wagner, “Untangling two systems of noncrossing curves,” Israel Journal of Mathematics, vol. 212, no. 1. Springer, pp. 37–79, 2016.
Matoušek J, Sedgwick E, Tancer M, Wagner U. 2016. Untangling two systems of noncrossing curves. Israel Journal of Mathematics. 212(1), 37–79.
Matoušek, Jiří, et al. “Untangling Two Systems of Noncrossing Curves.” Israel Journal of Mathematics, vol. 212, no. 1, Springer, 2016, pp. 37–79, doi:10.1007/s11856-016-1294-9.
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