Inflations of ideal triangulations

TL;DR

This paper introduces a method called inflation to transform ideal triangulations of 3-manifolds into more complete triangulations of the manifold itself, with practical algorithms and minimal examples.

Contribution

It presents a new inflation technique for ideal triangulations, including algorithms and minimal triangulation examples for specific 3-manifolds.

Findings

Provides an algorithm for inflation of ideal triangulations.

Constructs minimal triangulations of the figure-eight knot exterior.

Constructs minimal triangulation of the Gieseking manifold.

Abstract

Starting with an ideal triangulation of the interior of a compact 3-manifold M with boundary, no component of which is a 2-sphere, we provide a construction, called an inflation of the ideal triangulation, to obtain a strongly related triangulations of M itself. Besides a step-by-step algorithm for such a construction, we provide examples of an inflation of the two-tetrahedra ideal triangulation of the complement of the figure-eight knot in the 3-sphere, giving a minimal triangulation, having ten tetrahedra, of the figure-eight knot exterior. As another example, we provide an inflation of the one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven tetrahedra, of a nonorientable compact 3-manifold with Klein bottle boundary. Several applications of inflations are discussed.