Posted as arXiv:2109.14570.
Using a combination of techniques geometric, topological, and computational, we have resolved two longstanding open problems in the theory of hyperbolic three-manifolds. To wit,
- whether or not any orientable one-cusped hyperbolic three-manifolds besides the figure-eight knot complement admit more than eight exceptional fillings (no, they don’t); and
- whether or not the closed orientable hyperbolic three-manifolds of second and third smallest volume are those of second and third smallest volume in the SnapPea census (yes, they are).
Along the way we also showed that the figure-eight knot complement and its sister are the hyperbolic 3-manifolds with least maximal cusp volume.