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  • 标题:Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries
  • 本地全文:下载
  • 作者:Chambers, Erin Wolf ; Lazarus, Francis ; de Mesmay, Arnaud
  • 期刊名称:LIPIcs : Leibniz International Proceedings in Informatics
  • 电子版ISSN:1868-8969
  • 出版年度:2021
  • 卷号:189
  • 页码:23:1-23:16
  • DOI:10.4230/LIPIcs.SoCG.2021.23
  • 出版社:Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
  • 摘要: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 Î", there is a polynomial-time algorithm to decide if γ lies in the normal subgroup generated by components of Î" in the fundamental group of the surface after attaching the curves to a basepoint.
  • 关键词:3-manifolds; surfaces; low-dimensional topology; contractibility; compressed curves
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