A catalogue of orientable 3-manifolds triangulated by 30 coloured tetrahedra

TL;DR

This paper presents a comprehensive catalogue of orientable 3-manifolds triangulated by 30 coloured tetrahedra, using computational graph methods to classify and identify these manifolds via their combinatorial and topological structures.

Contribution

It introduces an automated approach to generate and classify orientable 3-manifolds with 30 tetrahedra using edge-coloured graphs and combinatorial moves, extending previous classifications.

Findings

Catalogue of all orientable 3-manifolds with 30 tetrahedra

Manifolds identified via JSJ decompositions and fibering structures

Crystallization equivalence classes correspond to homeomorphism classes

Abstract

The present paper follows the computational approach to 3-manifold classification via edge-coloured graphs, already performed by several authors with respect to orientable 3-manifolds up to 28 coloured tetrahedra, non-orientable 3-manifolds up to 26 coloured tetrahedra, genus two 3-manifolds up to 34 coloured tetrahedra: in fact, by automatic generation and analysis of suitable edge-coloured graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting coloured triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which are proved to be in one-to one correspondence with the homeomorphism classes of the represented manifolds.