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# Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

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LIPIcs.SoCG.2021.23.pdf
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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 [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.

## Cite As

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

## 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.

## Subject Classification

##### 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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