On the Complexity of Immersed Normal Surfaces

Benjamin A. Burton; Éric Colin de Verdière; Arnaud de Mesmay

doi:10.2140/gt.2016.20.1061

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arXiv · 1412.4988

On the Complexity of Immersed Normal Surfaces

Benjamin A. Burton·Éric Colin de Verdière·Arnaud de Mesmay

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Abstract

Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our proof uses a reduction from Boolean constraint satisfaction problems where every variable appears in at most two clauses, using a classification theorem of Feder. We also investigate variants, and provide a polynomial-time algorithm to test for a local version of this problem.

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Benjamin A. Burton, Éric Colin de Verdière, Arnaud de Mesmay. 2014-12-16. On the Complexity of Immersed Normal Surfaces. https://doi.org/10.2140/gt.2016.20.1061

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