Hyperbolic knot complements and commensurability, I

Two manifolds are commensurable if they share a common finite sheered cover. Partitioning the set of hyperbolic 3-manifolds into commensurability classes organizes this set along similar geometric lines. However, however the general problem is quite difficult. Instead we will focus on a manifolds that appear rare in a commensurability class. In fact, Reid and Walsh conjecture that there are at most three hyperbolic knot complements in commensurability class. The first talk will focus on the background and motivation for this conjecture. The second talk will focus on recent progress toward proving this conjecture.