Spacious knots

Abstract

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.

arXiv commentV3: 10 pages, 1 figure. Minor changes. To appear in Mathematical Research Letters. V2: 10 pages, 1 figure. Details added to proof of lemma 4.1, as well as minor revisions elsewhere. V1: 8 pages, 1 figure