Spacious knots

TL;DR

This paper constructs hyperbolic knots in the 3-sphere whose complements are predominantly composed of points with large injectivity radius, demonstrating a form of convergence to hyperbolic space.

Contribution

It proves the existence of knots with complements where the large injectivity radius region dominates volume, answering a question by Brock and Dunfield.

Findings

Existence of knots with volume concentrated in the thick part.

Knots can be constructed to approximate hyperbolic space in the Benjamini--Schramm sense.

Provides a sequence of knots converging to hyperbolic space as parameters vary.

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.