$3$-manifolds represented by $4$-regular graphs with three Eulerian cycles

TL;DR

This paper introduces a new class of hyperbolic 3-manifolds constructed from 4-regular graphs with Eulerian cycles, analyzing their complexity, triangulations, and enumeration bounds.

Contribution

It defines a novel construction of hyperbolic 3-manifolds using Eulerian cycles in graphs and establishes their properties and enumeration bounds.

Findings

Each manifold has Matveev complexity equal to n.

Unique minimal ideal triangulation with n tetrahedra.

Number of manifolds grows factorially with n, bounded between n! and n!4^n.

Abstract

We construct and study a new class $\mathscr{M}=\{\mathscr{M}_n\}_{n\ge 4}$ of compact hyperbolic $3$-manifolds with totally geodesic boundary. The members of $\mathscr{M}_n$ are defined via triples of pairwise compatible Eulerian cycles in $4$-regular $n$-vertex graphs. We show that each $M$ in $\mathscr{M}_n$ is of Matveev complexity $n$ and has a unique minimal ideal triangulation, which consists of $n$ tetrahedra. We exploit these properties to show that $n!\,4^n > |\mathscr{M}_n| > n!$ for each sufficiently large $n\in\mathbb{N}$.