Knot complements decomposing into prisms

The titular knot complements give counterexamples to two conjectures in the study of finite-volume hyperbolic 3-manifolds from the 1990’s. I’ll introduce these conjectures, on the non-existence of hyperbolic knot complements in S^3 with closed, embedded totally geodesic surfaces or with hidden symmetries (in the latter case, beyond a few already-known examples), and give an overview of the evidence in their favor that had accumulated until recently. Then I’ll describe the joint search with Arshia Gharagozlou and Neil Hoffman that turned up our new examples. I’ll close with further directions and still-open questions.