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Essential normal and spun normal surfaces in 3-manifolds


Authors: Ensil Kang and J. Hyam Rubinstein
Journal: Proc. Amer. Math. Soc. 146 (2018), 4967-4979
MSC (2010): Primary 57M99; Secondary 57M10
DOI: https://doi.org/10.1090/proc/14069
Published electronically: August 8, 2018
MathSciNet review: 3856162
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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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Additional Information

Ensil Kang
Affiliation: Department of Mathematics, College of Natural Sciences, Chosun University, Kwangju, 501-759, South Korea
Email: ekang@chosun.ac.kr

J. Hyam Rubinstein
Affiliation: School of Mathematics and Statistics, University of Melbourne Parkville, Peter Hall Building, Parkville, Victoria, Australia
Email: joachim@unimelb.edu.au

DOI: https://doi.org/10.1090/proc/14069
Keywords: 3-manifold, normal surface, triangulation, essential surface
Received by editor(s): September 23, 2015
Received by editor(s) in revised form: October 24, 2017, and December 27, 2017
Published electronically: August 8, 2018
Additional Notes: The first author was supported by research funds from Chosun University, 2014.
The second author was supported by the Australian Research Council grant DP13010369.
Communicated by: David Futer
Article copyright: © Copyright 2018 American Mathematical Society