Converting between quadrilateral and standard solution sets in normal surface theory

TL;DR

This paper introduces algorithms to convert between quadrilateral and standard solution sets in normal surface theory, significantly enhancing the efficiency of enumerating normal surfaces in 3-manifold topology.

Contribution

It presents a novel algorithm for converting between solution sets, combining the speed of quadrilateral coordinates with the comprehensiveness of standard coordinates.

Findings

Significantly faster enumeration of normal surfaces in practice

Speed improvements by factors up to millions in large cases

Effective implementation in the Regina software package

Abstract

The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.