Hidden symmetries and the Dehn fillings of all but one cusp of an infinite family of cyclic covers of a tetrahedral link complement

Priyadip Mondal (Rutgers)

Abstract: An isometry between two finite degree covers of a (finite volume) hyperbolic 3-manifold M is called a hidden symmetry of M if it is not a lift of a self-isometry of M. The existence of hidden symmetries in hyperbolic knot complements is an interesting and rare phenomenon - we are aware of only three hyperbolic knots whose complements have hidden symmetries. These knots are the figure eight knot and the two dodecahedral knots of Aitchison and Rubinstein. In 1992, Neumann and Reid asked if these three are the only hyperbolic knots whose complements have hidden symmetries. This question by Neumann and Reid has been central to the study of hidden symmetries in recent years. In our talk, we will study the existence of hidden symmetries in hyperbolic knot complements obtained by Dehn filling all but a single cusp of a hyperbolic link. In particular, the infinite family of link complements whose Dehn fillings we will pay attention to all cyclically cover a single hyperbolic link complement from the Fominykh-Garoufalidis-Goerner-Tarkaev-Vesnin tetrahedral census.