A new approach to crushing 3-manifold triangulations

TL;DR

This paper simplifies and generalizes the crushing operation in 3-manifold topology, making it more accessible and extending its application to non-orientable manifolds with practical algorithms.

Contribution

It introduces a new, simplified analysis of the crushing process, enabling its extension to non-orientable 3-manifolds and improving practical algorithms.

Findings

Simplified the analysis of the crushing operation into atomic steps

Extended crushing techniques to non-orientable 3-manifolds

Developed practical algorithms for non-orientable prime decomposition

Abstract

The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3-manifold topology: it enabled the first practical implementations of 3-sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in state-of-the-art algorithms for unknot recognition and testing for essential surfaces. Although the crushing operation will always reduce the size of a triangulation, it might alter its topology, and so it requires a careful theoretical analysis for the settings in which it is used. The aim of this short paper is to make the crushing operation more accessible to practitioners, and easier to generalise to new settings. When the crushing operation was first introduced, the analysis was powerful but extremely complex. Here we give a new treatment that reduces the crushing process to a sequential combination of three "atomic" operations on a cell decomposition, all of which are simple to analyse. As an application, we generalise the crushing operation to the setting of non-orientable 3-manifolds, where we obtain a new practical and robust algorithm for non-orientable prime decomposition. We also apply our crushing techniques to the study of non-orientable minimal triangulations.