Finding Non-Orientable Surfaces in 3-Manifolds

Authors Benjamin A. Burton, Arnaud de Mesmay, Uli Wagner

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Benjamin A. Burton
Arnaud de Mesmay
Uli Wagner

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Benjamin A. Burton, Arnaud de Mesmay, and Uli Wagner. Finding Non-Orientable Surfaces in 3-Manifolds. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.
  • 3-manifold
  • low-dimensional topology
  • embedding
  • non-orientability
  • normal surfaces


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