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Mathematics Seminar Date: Wednesday, 12 March 2025 Time: 3:30 PM Venue: Seminar Hall Hidden symmetries of hyperbolic knot complements Priyadip Mondal Ben Gurion University. 12-03-25 Abstract An isometry between two finite sheeted covers of a finite volume hyperbolic three-manifold M that is not a lift of any self-isometry of M is called a hidden symmetry of M. Arithmetic hyperbolic three-manifolds are known to admit (infinitely many) hidden symmetries. However, Reid proved that the figure eight knot is the only arithmetic hyperbolic knot. Moreover, our knowledge of examples of non-arithmetic hyperbolic knot complements with hidden symmetries is limited. The following Neumann and Reid question from 1992 put this more precisely: Are the two dodecahedral knots of Aitchison and Rubinstein and the figure eight knot the only hyperbolic knots whose complements have hidden symmetries? Our talk will be guided by the above question. Specifically, we will be interested in understanding when we can obtain an infinite family of knot complements with hidden symmetries by Dehn filling all but one component of a hyperbolic link which geometrically converge to that link complement. First, I will explain why we have this specific interest and discuss some results relating to this and some particular horoball packings corresponding to the original link complement. Finally, we will address which links are good candidates for our study of such Dehn fillings, and with this in mind, we will focus on such Dehn fillings of the links in the tetrahedral census of Fominykh, Garoufalidis, Goerner, Tarkaev, and Vesnin.
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