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 ${\mathbb{R}}^3$. 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 ${\mathbb{R}}^3$ or $\sp^3$ is generally less than $\frac{\pi }{\sqrt{12}}$, the density of the hexagonal packing of unit disks in ${\mathbb{R}}^2$.

6 pages; to appear in Contemporary Mathematics volume edited by Calvo, Millett & Rawdon