Spacious knots
Autumn E. Kent, Jessica S. Purcell

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 , 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 . As R approaches infinity, and as 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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Connective tissue disorders research
