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2016 On the complexity of immersed normal surfaces
Benjamin Burton, Éric Colin de Verdière, Arnaud de Mesmay
Geom. Topol. 20(2): 1061-1083 (2016). DOI: 10.2140/gt.2016.20.1061


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 Burton. Éric Colin de Verdière. Arnaud de Mesmay. "On the complexity of immersed normal surfaces." Geom. Topol. 20 (2) 1061 - 1083, 2016.


Received: 16 October 2014; Accepted: 28 June 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1339.68258
MathSciNet: MR3493099
Digital Object Identifier: 10.2140/gt.2016.20.1061

Primary: 57N10 , 68Q17
Secondary: 57Q35 , 68U05

Keywords: computational complexity , constraint satisfaction problem , immersed normal surface , low-dimensional topology , normal surface , three-manifold

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.20 • No. 2 • 2016
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