The Spectra of Volume and Determinant Densities of Links

TL;DR

This paper investigates the distribution of volume and determinant densities in hyperbolic links, showing they are dense in certain intervals and can approach the volume of an ideal hyperbolic octahedron.

Contribution

It demonstrates that sequences of non-alternating links can have volume densities approaching $v_8$, and the set of volume and determinant densities are dense in [0, v_8].

Findings

Volume densities of links can approach $v_8$.

Set of volume densities is dense in [0, v_8].

Set of determinant densities contains [0, v_8].

Abstract

The $\textit{volume density}$ of a hyperbolic link $K$ is defined to be the ratio of the hyperbolic volume of $K$ to the crossing number of $K$. We show that there are sequences of non-alternating links with volume density approaching $v_8$, where $v_8$ is the volume of the ideal hyperbolic octahedron. We show that the set of volume densities is dense in $[0,v_8]$. The $\textit{determinant density}$ of a link $K$ is $[2 \pi \log \det(K)]/c(K)$. We prove that the closure of the set of determinant densities contains the set $[0, v_8]$.