The Spectra of Volume and Determinant Densities of Links

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 $[0, v_8]$.