Maximal admissible faces and asymptotic bounds for the normal surface solution space

Burton, Benjamin A. (2011) Maximal admissible faces and asymptotic bounds for the normal surface solution space. Journal of Combinatorial Theory: Series A, 118 4: 1410-1435. doi:10.1016/j.jcta.2010.12.011

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Author Burton, Benjamin A.
Title Maximal admissible faces and asymptotic bounds for the normal surface solution space
Journal name Journal of Combinatorial Theory: Series A   Check publisher's open access policy
ISSN 0021-9800
Publication date 2011-05
Sub-type Article (original research)
DOI 10.1016/j.jcta.2010.12.011
Open Access Status
Volume 118
Issue 4
Start page 1410
End page 1435
Total pages 26
Place of publication Maryland Heights, MO, U.S.A.
Publisher Academic Press
Collection year 2012
Language eng
Formatted abstract
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system.

To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding “admissible faces”. Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.
Keyword 3-manifolds
Normal surfaces
Face lattice
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Official 2012 Collection
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Citation counts: TR Web of Science Citation Count  Cited 2 times in Thomson Reuters Web of Science Article | Citations
Scopus Citation Count Cited 4 times in Scopus Article | Citations
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Created: Tue, 28 Jun 2011, 10:57:03 EST by Dr Benjamin Burton on behalf of School of Mathematics & Physics