Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations

TL;DR

This paper introduces a method to efficiently prune the enumeration of 3-manifold triangulations by detecting genus in partial vertex links, significantly speeding up the process and enabling large-scale computations.

Contribution

The paper presents a novel technique for early detection of non-manifold triangulations via genus detection in vertex links, improving enumeration efficiency.

Findings

Achieved up to 249x speed-up in enumeration tasks.

Developed a method for logarithmic time boundary edge manipulation.

Enabled large-scale triangulation datasets using high-performance computing.

Abstract

Enumerating all 3-manifold triangulations of a given size is a difficult but increasingly important problem in computational topology. A key difficulty for enumeration algorithms is that most combinatorial triangulations must be discarded because they do not represent topological 3-manifolds. In this paper we show how to preempt bad triangulations by detecting genus in partially-constructed vertex links, allowing us to prune the enumeration tree substantially. The key idea is to manipulate the boundary edges surrounding partial vertex links using expected logarithmic time operations. Practical testing shows the resulting enumeration algorithm to be significantly faster, with up to 249x speed-ups even for small problems where comparisons are feasible. We also discuss parallelisation, and describe new data sets that have been obtained using high-performance computing facilities.