Abstract:
A hidden symmetry of a finite volume hyperbolic three-manifold M is an isometry between two finite degree covers of M that is not a lift of any self-isometry of M. Hyperbolic knot complements with hidden symmetries are of particular interest since, so far, we only know three of them: the figure eight knot and the two dodecahedral knots of Aitchison and Rubinstein. Indeed, Neumann and Reid asked in 1992 if these three knots exhaust the list of hyperbolic knots whose complements admit hidden symmetries.
This Neumann-Reid question will be the center of our talk. We will focus on families of knot complements obtained by Dehn filling all but one cusp of a link complement which geometrically converge to that said link complement. I will describe some techniques for studying such families. Finally, I will address why understanding such families obtained from links in the orientable tetrahedral census of Fominykh, Garoufalidis, Goerner, Tarkaev and Vesnin is important and explain how our techniques fit into this scenario.
