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

Abstract

We present explicit geometric decompositions of the hyperbolic complements of alternating kuniform tiling links, which are alternating links whose projection graphs are k-uniform tilings of S 2 , E 2 , or 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 H 2 . In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces Sg × I with genus g ≥ 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 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 ideal regular octahedron.

Hyperbolicity of links in thickened surfaces

“…By taking the quotient by an orientation-preserving subgroup of the symmetry group of the tiling, we obtain the projection of a fully alternating link on a closed orientable surface S of positive genus, where the link lives in S × I. This association of tilings to hyperbolic links has appeared previously for certain Euclidean uniform tilings [5] and for Euclidean and hyperbolic k-uniform tilings in [2]. By applying Theorem 1, we can now turn a broader class of tilings, with polygons not necessarily regular, into hyperbolic links in S × I.…”

“…Note that by adding bigon faces to 3-regular tilings, we can turn them into 4-regular tilings to obtain similar results. See [2] for more details and explicit calculations.…”

Geometry of biperiodic alternating links

“…The main result in this section is the following theorem. Part (1) was proved independently in [3]. (1) (T 2 × I) − L has a complete hyperbolic structure coming from a decomposition into regular ideal bipyramids on the faces of T L :…”

TG-Hyperbolicity of virtual links

“…Proof. This follows from Theorem 3.8 and Corollary 4.1 in [3], which states that the volume of a hyperbolic link L in S × I is bounded above by 2v oct c(L ).…”

“…So it is only necessary to show the virtual trefoil has the least volume among virtual knots of genus 1. It is conjectured that the minimally twisted chain link of three components 6 3 3 is the 3-cusped manifold of minimal volume. If true, this would imply the conjecture, since that link contains a sub-link that is a Hopf link, and hence the link complement is equivalent to a knot complement in T × I.…”