Constructing knots with specified geometric limits

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.

arXiv comment27 pages, 9 figures. V2: Improved exposition

References in corpus (7)

• Realising end invariants by limits of minimally parabolic, geometrically finite groups
• Effective bilipschitz bounds on drilling and filling
• Volumes of highly twisted knots and links
• Geometric limits of knot complements
• Geometry of planar surfaces and exceptional fillings
• Essential surfaces in highly twisted link complements
• Hyperbolicity of Augmented Links in the Thickened Torus