When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.23
URN: urn:nbn:de:0030-drops-138223
URL: https://drops.dagstuhl.de/opus/volltexte/2021/13822/
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### Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

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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.

### BibTeX - Entry

```@InProceedings{chambers_et_al:LIPIcs.SoCG.2021.23,
author =	{Chambers, Erin Wolf and Lazarus, Francis and de Mesmay, Arnaud and Parsa, Salman},
title =	{{Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries}},
booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
pages =	{23:1--23:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-184-9},
ISSN =	{1868-8969},
year =	{2021},
volume =	{189},
editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},