Counting gluings of octahedra

TL;DR

This paper develops a combinatorial framework for counting three-dimensional colored triangulations formed by gluing octahedra, characterizes those maximizing edges, and establishes a bijection with a family of trees for enumeration.

Contribution

It introduces a new approach to enumerate 3D triangulations via octahedral gluings and establishes a bijection with tree structures for exact counting.

Findings

Maximal edge gluings have the topology of the 3-sphere.

These gluings are in bijection with a family of trees.

Exact enumeration confirms the combinatorial results.

Abstract

Three--dimensional colored triangulations are gluings of tetrahedra whose faces carry the colors 0, 1, 2, 3 and in which the attaching maps between tetrahedra are defined using the colors. This framework makes it possible to generalize the notion of two--dimensional $2p$--angulations to three dimensions in a way which is suitable for combinatorics and enumeration. In particular, universality classes of three--dimensional triangulations can be investigated within this framework. Here we study colored triangulations obtained by gluing octahedra. Those which maximize the number of edges at fixed number of octahedra are fully characterized and are shown to have the topology of the 3--sphere. They are further shown to be in bijection with a family of trees, a result which is confirmed by the exact enumeration.