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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.23
URN: urn:nbn:de:0030-drops-138223
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13822/
Chambers, Erin Wolf ;
Lazarus, Francis ;
de Mesmay, Arnaud ;
Parsa, Salman
Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries
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},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13822},
URN = {urn:nbn:de:0030-drops-138223},
doi = {10.4230/LIPIcs.SoCG.2021.23},
annote = {Keywords: 3-manifolds, surfaces, low-dimensional topology, contractibility, compressed curves}
}
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Keywords: |
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3-manifolds, surfaces, low-dimensional topology, contractibility, compressed curves |
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Collection: |
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37th International Symposium on Computational Geometry (SoCG 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.06.2021 |
