K-normal surfaces

Evgeny Fominykh Institute of Mathematics and Mechanics Bruno Martelli Universit`a di Pisa

Differential Geometry mathscidoc:1609.10113

Journal of Differential Geometry, 82, (1), 101-114, 2009
Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) al- most normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: a minimal triangulation of a closed irreducible or a boundedhyperbolic 3-manifold contains no non-trivial k-normal sphere; every triangulation of a closed manifold with at least 2 tetra-hedra contains some non-trivial normal surface; every manifold with boundary has only finitely many triangulations without non-trivial normal surfaces. Here, triangulations of bounded manifolds are actually ideal triangulations. We also calculate the number of normal surfaces of nonnegative Euler characteristics which are contained in the conjecturally minimal triangulations of all lens spaces Lp,q.
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@inproceedings{evgeny2009k-normal,
  title={k-NORMAL SURFACES},
  author={Evgeny Fominykh, and Bruno Martelli},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911182605392030770},
  booktitle={Journal of Differential Geometry},
  volume={82},
  number={1},
  pages={101-114},
  year={2009},
}
Evgeny Fominykh, and Bruno Martelli. k-NORMAL SURFACES. 2009. Vol. 82. In Journal of Differential Geometry. pp.101-114. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911182605392030770.
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