2016/03/11 by Colin Adams, Adams, Colin, Aaron J. Calderon +11
Computer Science · Mathematics · #57M50 #Advanced Combinatorial Mathematics #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1603.03715
openalex publication_date 2016/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Sg 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 Sg x I, ranging over all g at least 2, is a dense subset of the interval [0, 2voct], where voct, approximately 3.66386, is the volume of the regular ideal octahedron.