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

TL;DR

This paper provides explicit geometric decompositions for hyperbolic complements of alternating $k$-uniform tiling links, revealing volume relationships and density results for hyperbolic links in thickened surfaces.

Contribution

It introduces a new geometric decomposition approach for these links and generalizes bipyramid constructions to analyze volume densities across surfaces of varying genus.

Findings

Spherical tiling links have volumes twice those of corresponding Archimedean solids.

Hyperbolic tiling links' structures derive from equilateral realizations of $k$-uniform tilings.

Volume densities for hyperbolic links in $S_g imes I$ are dense in [0, 2v_oct].

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 $S^2$, $\mathbb{E}^2$, or $\mathbb{H}^2$. 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 $\mathbb{H}^2$. In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces $S_g \times I$ with genus $g \ge 2$ 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 $S_g \times I$, ranging over all $g \ge 2$, is a dense subset of the interval $[0, 2v_\text{oct}]$, where $v_\text{oct} \approx 3.66386$ is the volume of the ideal regular octahedron.