Generalized Bipyramids and Hyperbolic Volumes of Tiling Links

We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of S2, E2, or H2. This requires generalizing the angle structures program of Casson and Rivin for triangulations with a mixture of finite, ideal, and truncated (i.e. ultra-ideal) vertices. A consequence of this decomposition is that the volumes of spherical tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description. In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces Sg×I with genus g ≥ 2. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for 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 regular ideal octahedron.