A hidden symmetry of a hyperbolic three-manifold M is an isometry between two finite-sheeted covers of M that is not the lift of any self-isometry of M. Arithmetic hyperbolic three-manifolds are known to have infinitely many hidden symmetries. Nonetheless, Alan Reid proved in 1991 that the figure-eight knot is the only hyperbolic knot whose complement is an arithmetic hyperbolic three-manifold. On the other hand, only a few non-arithmetic hyperbolic knot complements with hidden symmetries are currently known. Indeed, Neumann and Reid asked in 1992 if any other hyperbolic knot complement besides the figure-eight knot complement and the two dodecahedral knot complements of Aitchison and Rubinstein has hidden symmetries. Only very recently, DeBlois, Gharagozlou and Hoffman gave a positive answer to this question—they showed the existence of four more hyperbolic knot complements with hidden symmetries in addition to the three above.

Driven by the lack of examples of hyperbolic knot complements with hidden symmetries, the following conjecture was put forward in a joint work with Chesebro and DeBlois: Given v>0, there exists at most finitely many hyperbolic knot complements with hidden symmetries that have volume less than v. Thanks to Thurston’s Dehn surgery theorem, this conjecture prompts us to study when a family of hyperbolic knot complements obtained by Dehn filling all but one cusp of a hyperbolic link complement and geometrically converging to the link complement can have infinitely many members with hidden symmetries. In this talk, I will highlight a work that analyzes such families of hyperbolic knot complements when the original link complement belongs to the oriented tetrahedral census of Fominykh, Garoufalidis, Goerner, Tarkaev and Vesnin. This talk will feature a lot of pictures and won’t assume any prior knowledge of hyperbolic geometry.