Abstract
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.
Highlights
Since the foundational work of Haken [7] on unknot recognition, the past decades have witnessed a flurry of algorithms designed to solve decision problems in low-dimensional topology
Many low-dimensional problems can be seen as an instance of the following question, which encompasses the class of problems that normal surface theory has been designed to solve: Generic 3-manifold problem Input: A 3-manifold M
To be consistent with the literature in low-dimensional topology, we will describe 3-manifolds not with simplicial complexes, but with the looser concept of triangulations, which are defined as a collection of n abstract tetrahedra, all of whose 4n faces are glued together in pairs
Summary
Since the foundational work of Haken [7] on unknot recognition, the past decades have witnessed a flurry of algorithms designed to solve decision problems in low-dimensional topology. Our work complements these by investigating the complexity of finding non-orientable surfaces in the most general setting Another motivation for studying this question comes from the higher dimensional analogues of graph embeddings. The proof of Theorem 1 starts to the aforementioned one for 3-Manifold Knot Genus by Agol, Hass and Thurston: the idea is to encode an instance of One-in-Three SAT within the embeddability of a non-orientable surface inside a 2-complex. This complex is turned into a 3-manifold by a thickening step and a doubling step.
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