# Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

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.

## Authors

• 9 publications
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• ### Deciding contractibility of a non-simple curve on the boundary of a 3-manifold: A computational Loop Theorem

We present an algorithm for the following problem. Given a triangulated ...
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We prove the first polynomial bound on the number of monotonic homotopy ...
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• ### Fast and Simple Methods For Computing Control Points

The purpose of this paper is to present simple and fast methods for comp...
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• ### Segmenting a Surface Mesh into Pants Using Morse Theory

A pair of pants is a genus zero orientable surface with three boundary c...
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• ### Determining surfaces of revolution from their implicit equations

Results of number of geometric operations (often used in technical pract...
07/10/2014 ∙ by Jan Vršek, et al. ∙ 0

• ### 3D Registration of Curves and Surfaces using Local Differential Information

This article presents for the first time a global method for registering...
04/02/2018 ∙ by Carolina Raposo, et al. ∙ 0