The cusp density of a finite-volume hyperbolic manifold measures how densely the manifold can be packed with solid torus-shaped `cusps' around its boundary components. These cusps can be thought of as solid torus neighborhoods of the components of a hyperbolic link in its complement. Results on the packing of horospheres in hyperbolic 3-space show that the cusp density is bounded above by 8.53..., and in 2001 Colin Adams proved that the values of the cusp density invariant on 3-manifolds are dense in the interval [0, 8.53...]. One might wonder if the same is true of the possible cusp densities of hyperbolic knot complements, or other restricted classes of 3-manifolds.

I will introduce hyperbolic geometry, knots and links, and hyperbolic knots and links, then show how Adams' construction proves the same result for the cusp densities of hyperbolic link complements and discuss a simpler but weaker construction for one-cusped hyperbolic manifolds.