Constructing knots with specified geometric limits

Urs Fuchs, Jessica S. Purcell and John Stewart

Vol. 324 (2023), No. 1, 111–142

DOI: 10.2140/pjm.2023.324.111

Abstract

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the 3-sphere, then such knots can be taken to lie in the 3-sphere. However, their proof was nonconstructive; no examples were produced. In this paper, we give a constructive proof in the geometrically finite case. That is, given a geometrically finite, tame hyperbolic 3-manifold with one end, we build an explicit family of knots whose complements converge to it geometrically. Our knots lie in the (topological) double of the original manifold. The construction generalises the class of fully augmented links to a Kleinian groups setting.

Keywords

knot, geometric limit, circle packing, Kleinian groups, fully augmented link

Mathematical Subject Classification

Primary: 30F40, 57K10, 57K32

Secondary: 52C26

Milestones

Received: 10 February 2022

Revised: 1 May 2023

Accepted: 6 May 2023

Published: 22 June 2023