  We show that there exist hyperbolic knots in the 3-sphere such that the set
of points of large injectivity radius in the complement take up the bulk of the
volume. More precisely, given a finite volume hyperbolic manifold, for any
bound R>0 on injectivity radius, consider the set of points with injectivity
radius at least R; we call this the R-thick part of the manifold. We show that
for any $\epsilon>0$, there exists a knot K in the 3-sphere so that the ratio
of the volume of the R-thick part of the knot complement to the volume of the
knot complement is at least $1-\epsilon$. As R approaches infinity, and as
$\epsilon$ approaches zero, this gives a sequence of knots that is said to
Benjamini--Schramm converge to hyperbolic space. This answers a question of
Brock and Dunfield.
