Spacious knots

Autumn E. Kent, Jessica S. Purcell

Research output: Contribution to journalArticleResearchpeer-review

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 > 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 − . As R approaches infinity, and as approaches 0, this gives a sequence of knots that is said to Benjamini–Schramm converge to hyperbolic space. This answers a question of Brock and Dunfield.

Original languageEnglish
Pages (from-to)581-595
Number of pages15
JournalMathematical Research Letters
Volume25
Issue number2
DOIs
Publication statusPublished - 1 Jan 2018

Cite this