On Thickness and Packing Density for Knots and Links

Abstract

We describe some problems, observations, and conjectures concerning thickness and packing density of knots and links in $\sp^3$ and $\R^3$. We prove the thickness of a nontrivial knot or link in $\sp^3$ is no more than $\frac{\pi}{4}$, the thickness of a Hopf link. We also give arguments and evidence supporting the conjecture that the packing density of thick links in $\R^3$ or $\sp^3$ is generally less than $\frac{\pi}{\sqrt{12}}$, the density of the hexagonal packing of unit disks in $\R^2$.