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

TL;DR

This paper improves asymptotic bounds on the number of admissible vertices in polytopes used for enumerating normal surfaces, which are crucial in computational 3D topology, by analyzing admissible faces in different coordinate systems.

Contribution

It introduces new bounds on maximal admissible faces and establishes a bijection between face structures in two coordinate systems, advancing understanding of the solution space structure.

Findings

Significant improvements on asymptotic bounds for admissible vertices.

Upper bounds on the number of maximal admissible faces.

Bijection between face lattices in different coordinate systems.

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.