Essential normal and spun normal surfaces in 3-manifolds

Kang, Ensil; Rubinstein, J.

doi:10.1090/proc/14069

Essential normal and spun normal surfaces in 3-manifolds
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by Ensil Kang and J. Hyam Rubinstein PDF

Proc. Amer. Math. Soc. 146 (2018), 4967-4979 Request permission

Abstract:

Normal and spun normal surfaces are key tools for algorithms in 3-dimensional geometry and topology, especially concerning essential surfaces. In a recent paper of Dunfield and Garoufalidis, an interesting criterion is given for a spun normal surface to be essential in an ideal triangulation of a 3-manifold with a complete hyperbolic metric of finite volume. Their method uses ideal points of character varieties and Culler–Shalen theory. In this paper, we give a simple proof of a criterion which applies for both triangulations of closed 3-manifolds and ideal triangulations of the interior of compact 3-manifolds, giving a sufficient condition for a normal or a spun normal surface to be essential. Our criterion implies that of Dunfield and Garoufalidis. We also give a necessary and sufficient condition for a normal surface in a closed 3-manifold to be essential, using sweepouts and almost normal surface theory.

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