Poor Ideal Three-Edge Triangulations Are Minimal

An ideal triangulation of a compact $ 3 $ -manifold with nonempty boundary is known to be minimal if and only if the triangulation contains the minimum number of edges among all ideal triangulations of the manifold. Therefore, every ideal one-edge triangulation (i.e., an ideal singular triangulation with exactly one edge) is minimal. Vesnin, Turaev, and Fominykh showed that an ideal two-edge triangulation is minimal if no $ 3 $ – $ 2 $ Pachner move can be applied. In this paper we show that each 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.