Constructing Knots with Specified Geometric Limits

Explore the intricacies of knot theory and hyperbolic geometry in this 52-minute lecture by Jessica Purcell from Monash University. Delve into the constructive proof of how infinite volume hyperbolic 3-manifolds with finitely generated fundamental groups can be geometric limits of knot complements. Discover the innovative use of circle packings on conformal boundaries of tame hyperbolic 3-manifolds to construct knots in the double of the original manifold. Examine the challenges of extending this construction to the 3-sphere and the additional properties of circle packings required. Gain insights into open questions and future research directions in this fascinating area of mathematics. The lecture, part of IPAM's Statistical Mechanics and Discrete Geometry Workshop at UCLA, presents joint work with Urs Fuchs and John Stewart, offering a deep dive into advanced topics in geometric topology.