Constructing knots with specified geometric limits

Jessica Purcell
Monash University

Any infinite volume hyperbolic 3-manifold with finitely generated fundamental group that embeds in the 3-sphere is known to be a geometric limit of knot complements, but the original proof was only existential -- no knots were constructed. Using circle packings on the conformal boundaries of tame hyperbolic 3-manifolds, we give a constructive proof of this result -- except we can only guarantee our knots lie in the double of the original manifold, and not in the 3-sphere. Extending to the 3-sphere will require additional properties of circle packings. In this talk, I will discuss the construction and related open questions. This is joint work with Urs Fuchs and John Stewart.

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