Topology and Geometric Group Theory Seminar
Given a hyperbolic knot K, the corresponding knot complement M has a number of interesting geometric invariants. Here, we shall consider the volume of M and the length spectrum of M, which is the set of all lengths of closed geodesics in M counted with multiplicities. It is natural to ask how bad are these invariants at distinguishing hyperbolic 3-manifolds and how do these invariants interact with one another. In this talk, we shall construct large families of hyperbolic pretzel knot complements with the same volume and the same initial length spectrum. This construction will rely on mutating pretzel knots along four-punctured spheres, and then showing that such mutations often preserve the volume and short geodesic lengths of a hyperbolic knot complement.