Hyperbolic graphs of small complexity

TL;DR

This paper classifies and enumerates the simplest pairs of 3-manifolds and embedded trivalent graphs using hyperbolic geometry, providing a comprehensive list and detailed invariants for pairs up to complexity 5.

Contribution

It introduces a classification of minimal 3-manifold-graph pairs based on hyperbolic structures, extending previous work with new computational and geometric methods.

Findings

129 pairs classified up to complexity 5

123 pairs are hyperbolic with parabolic meridians

Detailed hyperbolic invariants provided for these pairs

Abstract

In this paper we enumerate and classify the ``simplest'' pairs (M,G) where M is a closed orientable 3-manifold and G is a trivalent graph embedded in M. To enumerate the pairs we use a variation of Matveev's definition of complexity for 3-manifolds, and we consider only (0,1,2)-irreducible pairs, namely pairs (M,G) such that any 2-sphere in M intersecting G transversely in at most 2 points bounds a ball in M either disjoint from G or intersecting G in an unknotted arc. To classify the pairs our main tools are geometric invariants defined using hyperbolic geometry. In most cases, the graph complement admits a unique hyperbolic structure with parabolic meridians; this structure was computed and studied using Heard's program Orb and Goodman's program Snap. We determine all (0,1,2)-irreducible pairs up to complexity 5, allowing disconnected graphs but forbidding components without vertices in complexity 5. The result is a list of 129 pairs, of which 123 are hyperbolic with parabolic meridians. For these pairs we give detailed information on hyperbolic invariants including volumes, symmetry groups and arithmetic invariants. Pictures of all hyperbolic graphs up to complexity 4 are provided. We also include a partial analysis of knots and links. The theoretical framework underlying the paper is twofold, being based on Matveev's theory of spines and on Thurston's idea (later developed by several authors) of constructing hyperbolic structures via triangulations. Many of our results were obtained (or suggested) by computer investigations.