  We present explicit geometric decompositions of the complement of tiling
links, which are alternating links whose projection graphs are uniform tilings
of the 2-sphere, the Euclidean plane or the hyperbolic plane. 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 S_g x I with genus g at least 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 S_g x I, ranging over all g at
least 2, is a dense subset of the interval [0, 2v_{oct}], where v_{oct},
approximately 3.66386, is the volume of the regular ideal octahedron.
