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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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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


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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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)


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.

Subject Classification

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


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