Computational topology with Regina: Algorithms, heuristics and implementations

TL;DR

Regina is a comprehensive software tool for 3-manifold topology that integrates algorithms, heuristics, and implementations, facilitating research and education in the field.

Contribution

This paper documents the key algorithms, heuristics, and data structures in Regina, providing the first detailed literature account of its core computational methods.

Findings

Implementation of efficient simplification heuristics

Modern algorithms for 3-sphere recognition and connected sum decomposition

Performance improvements in normal surface enumeration

Abstract

Regina is a software package for studying 3-manifold triangulations and normal surfaces. It includes a graphical user interface and Python bindings, and also supports angle structures, census enumeration, combinatorial recognition of triangulations, and high-level functions such as 3-sphere recognition, unknot recognition and connected sum decomposition. This paper brings 3-manifold topologists up-to-date with Regina as it appears today, and documents for the first time in the literature some of the key algorithms, heuristics and implementations that are central to Regina's performance. These include the all-important simplification heuristics, key choices of data structures and algorithms to alleviate bottlenecks in normal surface enumeration, modern implementations of 3-sphere recognition and connected sum decomposition, and more. We also give some historical background for the project, including the key role played by Rubinstein in its genesis 15 years ago, and discuss current directions for future development.