Poor ideal three-edge triangulations are minimal

TL;DR

The paper proves that certain 'poor' ideal three-edge triangulations are minimal and uses this to construct minimal triangulations for an infinite family of hyperbolic 3-manifolds with totally geodesic boundary.

Contribution

It establishes the minimality of poor ideal three-edge triangulations and applies this to generate minimal triangulations for an infinite class of hyperbolic 3-manifolds.

Findings

Poor ideal three-edge triangulations are minimal.

Minimal triangulations can be constructed for an infinite family of hyperbolic 3-manifolds.

Provides a criterion for minimality based on triangulation properties.

Abstract

It is known that an ideal triangulation of a compact $3$-manifold with nonempty boundary is minimal if and only if it contains the minimum number of edges among all ideal triangulations of the manifold. Therefore, any ideal one-edge triangulation (i.e., an ideal singular triangulation with exactly one edge) is minimal. Vesnin, Turaev, and the first author showed that an ideal two-edge triangulation is minimal if no $3$-$2$ Pachner move can be applied. In this paper we show that any of the so-called poor ideal three-edge triangulations is minimal. We exploit this property to construct minimal ideal triangulations for an infinite family of hyperbolic $3$-manifolds with totally geodesic boundary.