Ideal simplices and double-simplices, their non-orientable hyperbolic manifolds, cone manifolds and orbifolds with Dehn type surgeries and graphic analysis
Let \(\mathcal{P}\) be a collection of ideal hyperbolic simplices in hyperbolic 3-dimensional space \(\mathbb{H}^3\), then by pairing and gluing the faces of \(\mathcal{P}\) we can obtain a cusped 3-manifold, which under certain conditions can be an hyperbolic 3-manifold. This paper is partly a survey of hyperbolic manifolds obtained by gluing in pairs the faces of an ideal hyperbolic 3-simplex \(\mathcal{S}\) or a pair of 3-simplices \(\mathcal{S}_1\), \(\mathcal{S}_2\). The Gieseking manifold is obtained by identifying in pairs the faces of an ideal 3-simplex; the result is a non-orientable hyperbolic 3-manifold \(M\), which is known to be the smallest volume non-compact hyperbolic 3-manifold. By means of a computational algorithm, the authors have determined all equivariance classes for combinatorially face pairings of a pair of simplices, and there are 5079 of these classes. Of these configurations, only seven are hyperbolic manifolds. The manifolds obtained are the complement of the figure eight knot (also known as Thurston manifold), a manifold which by Dehn surgery produces the Fomenko-Matveev-Weeks manifold (which is know to be the smallest volume closed orientable hyperbolic 3-manifold), and a 2-cusped non-orientable manifold \(M_{c^2}\), which has been considered by several authors, and in fact this manifold is represented by 5 non-equivariant face pairings. Then the authors consider the Dehn type surgeries of \(M_{c^2}\) that produce compact hyperbolic cone manifolds, especially orbifolds. As a warm up, Dehn type surgeries on the Gieseking manifold are first considered. A series of tables is given, showing the calculation of the parameters and volume for the cone manifolds considered.
hyperbolic manifold and cone manifold by fundamental polyhedron
scientific article; zbMATH DE number 7362843