Horoball packing, hidden symmetry and some tetrahedral links

Given a hyperbolic link in $S^3$, one can associate certain packings of the hyperbolic $3$-space called horoball packings. This talk aims to discern the role of horoball packings in the context of hidden symmetries.  For a hyperbolic knot $K$, we say that an isometry $g$ between two finite degree covers of $S^3-K$ is a hidden symmetry of $S^3-K$ if $g$ is not a lift of a self-isometry of $S^3-K$. In 1992, Neumann and Reid asked whether the figure-eight knot and the two dodecahedral knots of Aitchison and Rubinstein are the only hyperbolic knots whose complements have hidden symmetries. The study of hidden symmetries in hyperbolic knot complements is primarily motivated by this question of Neumann and Reid. Our focus in the talk will be the study of the existence of hidden symmetries for hyperbolic knots obtained by Dehn filling all but one component of a given hyperbolic link $L$. I will explain how the understanding of certain horoball packings of $H^3$ associated with $L$ (and the related circle packings of the complex plane) guides us in that pursuit. Finally, I will discuss the application of this strategy for some links in the tetrahedral census of Fominykh-Garoufalidis-Goerner-Tarkaev-Vesnin.