Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions

Abstract

We study combinatorial properties of polytopes realizable in the Lobachevsky space \(\mathbb{L}^{3}\) as polytopes of finite volume with right dihedral angles. On the basis of E. M. An-dreev’s theorem we prove that cutting off ideal vertices of right-angled polytopes defines a one-to-one correspondence with strongly cyclically four-edge-connected polytopes different from the cube and the pentagonal prism. We show that any polytope of the latter family can be obtained by cutting off a matching of a polytope from the same family or of the cube with at most two nonadjacent orthogonal edges cut, in such a way that each quadrangle results from cutting off an edge. We refine D. Barnette’s construction of this family of polytopes and present its application to right-angled polytopes. We refine the known construction of ideal right-angled polytopes using edge twists and describe its connection with D. Barnette’s construction via perfect matchings. We make a conjecture on the behavior of the volume under operations and give arguments to support it.