Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links

Abstract

We present explicit geometric decompositions of the hyperbolic complements of alternating k-uniform tiling links, which are alternating links whose projection graphs are k-uniform tilings of S2, E2, or H2. A consequence of this decomposition is that the volumes of spherical alternating k-uniform tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description, and the hyperbolic structures for the hyperbolic alternating tiling links come from the equilateral realization of the k-uniform tiling on H2. In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces Sg×I with genus g≥2 and totally geodesic boundaries. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for all hyperbolic links in Sg×I, ranging over all g≥2, is a dense subset of the interval [0,2voct], where voct≈3.66386 is the volume of the ideal regular octahedron.