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Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries


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 size 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 \(\gamma \), and a collection of disjoint normal curves \(\varDelta \), there is a polynomial-time algorithm to decide if \(\gamma \) lies in the normal subgroup generated by components of \(\varDelta \) in the fundamental group of the surface after attaching the curves to a basepoint.

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  1. There are two different forms of compression going on in this paper: normal coordinates and straight line programs. We keep the name compressed for the latter, while a curve or a multicurve encoded with normal coordinates will be simply called a normal (multi-)curve.

  2. We recall, however, that the homeomorphism problem is undecidable in dimensions four or higher.

  3. A wedge of a family of topological spaces is the space obtained after attaching them all to a single common point. Actually, there are also circle summands here—we do not mention them in this outline to keep the discussion light.

  4. Their normal coordinates are given by the restriction on S of those of \(\varDelta \).

  5. At the start of Steps 1, 2, and 3, it might happen that the last letter of \(A_j^R\) and the first letter of \(A_k^R\) are both edges, if half of a jumping turn got removed. In that case, we remove those edges since they are superfluous.


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The authors would like to thank Mark Bell, Ben Burton, and Jeff Erickson for helpful discussions. We also thank an anonymous referee of [9] (which is a heavily modified version of [8]) 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. We are also grateful to the anonymous reviewers for their careful reading of the manuscript which greatly improved the paper.

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Correspondence to Francis Lazarus.

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Erin Wolf Chambers: This work was funded in part by the National Science Foundation through grants CCF-1614562, CCF-1907612 and DBI-1759807.

Francis Lazarus: 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.

Arnaud de Mesmay: 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).

Salman Parsa: This work was funded in part by the National Science Foundation through grant CCF-1614562 as well as funding from the SLU Research Institute.

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Chambers, E.W., Lazarus, F., de Mesmay, A. et al. Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries. Discrete Comput Geom (2022).

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  • 3-Manifolds
  • Surfaces
  • Computational topology
  • Contractibility
  • Compressed curves

Mathematics Subject Classification

  • 57M05
  • 68R99