Constructing knots with specified geometric limits

TL;DR

This paper provides a constructive method to generate explicit knots in the 3-sphere whose complements approximate given geometrically finite hyperbolic 3-manifolds, extending previous nonconstructive results.

Contribution

It introduces a constructive approach to realize specific hyperbolic 3-manifolds as limits of knot complements, generalizing augmented links to Kleinian groups.

Findings

Explicit family of knots constructed for given manifolds

Knots lie in the double of the original manifold

Convergence of knot complements to target manifold

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.