Seminars & Colloquia Calendar

Abstract: Given a hyperbolic 3-manifold M, an isometry g between two finite sheeted covers of M is said to be a hidden symmetry of M if g is not a lift of any self-isometry of M. This talk will focus on hidden symmetries of hyperbolic knot complements. The interest behind this comes from a question of Neumann and Reid from 1992, which asks whether there are (hyperbolic) knot complements other than the figure-eight knot complement and the complements of the two dodecahedral knots of Aitchison and Rubinstein which have hidden symmetries.

In particular, we restrict our attention to the families of hyperbolic knots obtained from Dehn filling all but a fixed cusp of a given hyperbolic link L. By analyzing different horoball packings of the hyperbolic 3-space coming from L, we investigate whether such a family when it “geometrically converges” to L can have infinitely many elements with hidden symmetries. We will also discuss some applications of this analysis to some links in the tetrahedral census provided by Fominykh, Garoufalidis, Goerner, Tarkaev, and Vesnin.