Mathematics – Geometric Topology
Scientific paper
2010-04-15
Journal of Combinatorial Theory, Series A 118 (2011), no. 4, 1410-1435
Mathematics
Geometric Topology
31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in Journal of Combinatorial Theory A)
Scientific paper
10.1016/j.jcta.2010.12.011
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.
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