Abstract
In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3manifold 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 (noncompressed) curves, and, in very limited cases, for curves with selfintersections. Furthermore, our algorithm is fixedparameter tractable in the complexity of the input 3manifold.
As part of our proof, we obtain new polynomialtime algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomialtime 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 polynomialtime 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 3Manifold Boundaries}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {23:123:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771849},
ISSN = {18688969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13822},
URN = {urn:nbn:de:0030drops138223},
doi = {10.4230/LIPIcs.SoCG.2021.23},
annote = {Keywords: 3manifolds, surfaces, lowdimensional topology, contractibility, compressed curves}
}
Keywords: 

3manifolds, surfaces, lowdimensional topology, contractibility, compressed curves 
Collection: 

37th International Symposium on Computational Geometry (SoCG 2021) 
Issue Date: 

2021 
Date of publication: 

02.06.2021 