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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
Authors
Urs Fuchs
RWTH Aachen University
Aachen
Germany
Jessica S. Purcell
School of Mathematics
Monash University
Melbourne
Australia
John Stewart
School of Mathematics and Statistics
The University of Melbourne
Parkville
Australia

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