Triangulations of 3-manifolds with essential edges

TL;DR

This paper introduces essential and strongly essential triangulations of 3-manifolds, providing multiple construction methods, conditions for their existence, and algorithms for their recognition, advancing understanding of 3-manifold topology and geometry.

Contribution

It defines essential and strongly essential triangulations, offers four construction techniques, and develops algorithms for their detection, linking geometric structures to topological properties.

Findings

Semi-angle structures ensure a triangulation is essential.

Strict angle structures guarantee a triangulation is strongly essential.

Algorithms are provided to test essentiality of 3-manifold triangulations.

Abstract

We define essential and strongly essential triangulations of 3-manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold. We also show that a semi-angle structure is a sufficient condition for a triangulation of a 3-manifold to be essential, and a strict angle structure is a sufficient condition for a triangulation to be strongly essential. Moreover, algorithms to test whether a triangulation of a 3-manifold is essential or strongly essential are given.