Spacious knots

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 $R \gt 0$ 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 $\varepsilon \gt 0$, 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 $1-\varepsilon$. As $R$ approaches infinity, and as $\varepsilon$ 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.

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The first author was supported by an NSF CAREER grant DMS-1350075. The second author was supported by NSF grant DMS-1252687 and a grant from the Australian Research Council. Both authors were supported by von Neumann Fellowships at the Institute for Advanced Study: This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors thank the referee for many useful comments that greatly improved the exposition.

Received 1 November 2016

Accepted 26 May 2017

Published 5 July 2018