The complexity of the normal surface solution space

TL;DR

This paper investigates the growth of vertex normal surfaces in 3D topology, providing tighter theoretical bounds, constructing worst-case examples, and analyzing millions of triangulations to understand practical behavior.

Contribution

It improves theoretical bounds on the number of normal surfaces and offers the first large-scale empirical analysis of their distribution in triangulations.

Findings

Theoretical exponential upper bounds are significantly tightened.

Constructed triangulations demonstrate exponential growth is unavoidable.

Empirical analysis shows the number of vertex normal surfaces is generally small.

Abstract

Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number of such surfaces grows in relation to the size of the underlying triangulation. Here we address this problem in both theory and practice. In theory, we tighten the exponential upper bound substantially; furthermore, we construct pathological triangulations that prove an exponential bound to be unavoidable. In practice, we undertake a comprehensive analysis of millions of triangulations and find that in general the number of vertex normal surfaces is remarkably small, with strong evidence that our pathological triangulations may in fact be the worst case scenarios. This analysis is the first of its kind, and the striking behaviour that we observe has important implications for the feasibility of topological algorithms in three dimensions.