The Pachner graph and the simplification of 3-sphere triangulations

TL;DR

This paper presents experimental evidence that simplifying 3-sphere triangulations requires minimal modifications, challenging the theoretical expectation of exponential complexity, with implications for topology algorithms.

Contribution

It introduces novel experimental methods showing that 3-sphere triangulations can be simplified with minimal additions and modifications, contrary to prior theoretical assumptions.

Findings

Never need to add more than two tetrahedra in simplification

Minimal local modifications suffice for 3-sphere triangulations

Experimental results challenge exponential complexity expectations

Abstract

It is important to have fast and effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental work suggesting that for 3-sphere triangulations the reality is far different: we never need to add more than two tetrahedra, and we never need more than a handful of local modifications. If true in general, these extremely surprising results would have significant implications for decision algorithms and the study of triangulations in 3-manifold topology. The algorithms behind these experiments are interesting in their own right. Key techniques include the isomorph-free generation of all 3-manifold triangulations of a given size, polynomial-time computable signatures that identify triangulations uniquely up to isomorphism, and parallel algorithms for studying finite level sets in the infinite Pachner graph.