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Untangling Two Systems of Noncrossing Curves

  • Jiří Matoušek
  • Eric Sedgwick
  • Martin Tancer
  • Uli Wagner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

Abstract

We consider two systems (α 1,…,α m ) and (β 1,…,β n ) of curves drawn on a compact two-dimensional surface \(\mathcal{M}\) with boundary. Each α i and each β j is either an arc meeting the boundary of \(\mathcal{M}\) at its two endpoints, or a closed curve. The α i are pairwise disjoint except for possibly sharing endpoints, and similarly for the β j . We want to “untangle” the β j from the α i by a self-homeomorphism of \(\mathcal{M}\); more precisely, we seek an homeomorphism \(\varphi\:\mathcal{M}\to\mathcal{M}\) fixing the boundary of \(\mathcal{M}\) pointwise such that the total number of crossings of the α i with the ϕ(β j ) 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 \(\mathcal{M}\) 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 \(\mathcal{M}\) 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.

Keywords

Curves on 2-manifolds simultaneous planar drawings Lickorish’s theorem 

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Jiří Matoušek
    • 1
    • 2
  • Eric Sedgwick
    • 3
  • Martin Tancer
    • 1
    • 4
  • Uli Wagner
    • 5
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic
  2. 2.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland
  3. 3.School of CTIDePaul UniversityChicagoUSA
  4. 4.Institutionen för matematikKTHStockholmSweden
  5. 5.IST AustriaKlosterneuburgAustria

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