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Article Dans Une Revue Discrete and Computational Geometry Année : 2022

Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

Résumé

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 size 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.

Dates et versions

hal-03835041 , version 1 (31-10-2022)

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Erin Wolf Chambers, Francis Lazarus, Arnaud de Mesmay, Salman Parsa. Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries. Discrete and Computational Geometry, 2022, ⟨10.1007/s00454-022-00411-x⟩. ⟨hal-03835041⟩
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