Generalized bipyramids and hyperbolic volumes of tiling links

TL;DR

This paper develops explicit geometric decompositions for tiling link complements, generalizing angle structures and bipyramid constructions, to analyze volumes in spherical and hyperbolic settings, revealing dense volume density ranges.

Contribution

It introduces a generalized angle structures framework and bipyramid constructions for tiling links, extending volume analysis to spherical and hyperbolic cases.

Findings

Volumes of spherical tiling links are twice the maximal volumes of corresponding Archimedean solids.

The set of volume densities for hyperbolic tiling links in thickened surfaces is dense in [0, 2v_{oct}].

Generalized bipyramids enable volume bounds for links in surfaces of genus g ≥ 2.

Abstract

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.