Optimizing the double description method for normal surface enumeration

TL;DR

This paper improves the efficiency of normal surface enumeration in 3-manifold topology by optimizing the double description method, resulting in faster algorithms with lower memory use, tested in the Regina software.

Contribution

It introduces new optimizations for the double description method, significantly enhancing performance in normal surface enumeration.

Findings

Reduced running time in vertex enumeration

Lower memory consumption during computations

Effective implementation in Regina software

Abstract

Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an account of the vertex enumeration problem as it applies to normal surfaces, and present new optimizations that yield strong improvements in both running time and memory consumption. The resulting algorithms are tested using the freely available software package Regina.