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Spiraling and Folding: The Topological View

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Abstract

For every n, we construct two arcs in the plane that intersect at least n times and do not form spirals. The construction is in three stages: we first exhibit two closed curves on the torus that do not form double spirals, then two arcs on the torus that do not form spirals, and finally two arcs in the plane that do not form spirals. The planar arcs provide a counterexample to a proof of Pach and Tóth concerning string graphs.

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Notes

  1. The problem is decidable on arbitrary surfaces, as shown in [10].

  2. We refer the reader to [3, 4, 16] for a basic reference on the topology of surfaces.

  3. Two curves are homotopic if they can be continuously deformed into each other.

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Author notes

  1. Supported by Project 22-19073 S of the Czech Science Foundation (GAČR) and by Charles University Project UNCE/SCI/004.

Authors and Affiliations

  1. Charles University, Prague, Czech Republic

    Jan Kynčl

  2. DePaul University, Chicago, IL, 60604, USA

    Marcus Schaefer & Eric Sedgwick

  3. University of Rochester, Rochester, NY, 14627, USA

    Daniel Štefankovič

Authors
  1. Jan Kynčl
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  2. Marcus Schaefer
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  3. Eric Sedgwick
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  4. Daniel Štefankovič
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Correspondence to Marcus Schaefer.

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An earlier version of this paper appeared in the proceedings of the 19th Canadian Conference on Computational Geometry [12]

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Kynčl, J., Schaefer, M., Sedgwick, E. et al. Spiraling and Folding: The Topological View. Discrete Comput Geom 72, 246–268 (2024). https://doi.org/10.1007/s00454-023-00603-z

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