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Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

Authors Erin Wolf Chambers, Francis Lazarus, Arnaud de Mesmay, Salman Parsa

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ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Mathematics of computing → Graphs and surfaces
  • Theory of computation → Data compression
Keywords
  • 3-manifolds
  • surfaces
  • low-dimensional topology
  • contractibility
  • compressed curves

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Abstract

In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold.
As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve γ, and a collection of disjoint normal curves Δ, there is a polynomial-time algorithm to decide if γ lies in the normal subgroup generated by components of Δ in the fundamental group of the surface after attaching the curves to a basepoint.

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Erin Wolf Chambers, Francis Lazarus, Arnaud de Mesmay, and Salman Parsa. Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.23

Author Details

Erin Wolf Chambers
  • St. Louis University, MO, USA
Francis Lazarus
  • G-SCOP, CNRS, UGA, Grenoble, France
Arnaud de Mesmay
  • LIGM, CNRS, Université Gustave Eiffel, ESIEE Paris, 77454 Marne-la-Vallée, France
Salman Parsa
  • St. Louis University, MO, USA

Funding

  • Chambers, Erin Wolf: This work was funded in part by the National Science Foundation through grants CCF-1614562, CCF-1907612 and DBI-1759807.
  • Lazarus, Francis: This author is partially supported by the French ANR projects GATO (ANR-16-CE40-0009-01) and MINMAX (ANR-19-CE40-0014) and the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d’avenir.
  • de Mesmay, Arnaud: This author is partially supported by the French ANR projects ANR-17-CE40-0033 (SoS), ANR-16-CE40-0009-01 (GATO) and ANR-19-CE40-0014 (MINMAX).
  • Parsa, Salman: This work was funded in part by the National Science Foundation through grant CCF-1614562 as well as funding from the SLU Research Institute.

Acknowledgements

The authors would like to thank Mark Bell, Ben Burton, and Jeff Erickson for helpful discussions. We also thank an anonymous referee of [de Verdi{è}re and Parsa, 2020] for suggesting the use of a maximal compression body as a simple exponential-time algorithm for deciding contractibility of arbitrary (non-compressed) curves on the boundary of a 3-manifold.

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