Videos

Recent Progress in Topological and Geometric Structures in Low Dimensions: Knot complements decomposing into prisms

Presenter
March 24, 2026
Keywords:
  • Surfaces
  • geometric 3-manifolds
  • mapping class groups
  • foliations
  • fibrations
  • Anosov flows
  • Teichmuller space
  • conformal structures
MSC:
  • 57M50 - General geometric structures on low-dimensional manifolds
Abstract
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.