Constructing knots with specified geometric limits

Urs Fuchs, Jessica S. Purcell, John Stewart

Research output: Contribution to journalArticleResearchpeer-review

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.

Original languageEnglish
Pages (from-to)111-142
Number of pages32
JournalPacific Journal of Mathematics
Volume324
Issue number1
DOIs
Publication statusPublished - 22 Jun 2023

Keywords

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

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