## 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 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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## Notes

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

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.Their normal coordinates are given by the restriction on

*S*of those of \(\varDelta \).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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## Acknowledgements

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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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). https://doi.org/10.1007/s00454-022-00411-x

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DOI: https://doi.org/10.1007/s00454-022-00411-x

### Keywords

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

### Mathematics Subject Classification

- 57M05
- 68R99