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Thursday, December 17, 2015

**Abstract:** Two 3-manifolds are commensurable if they share a common finite sheeted cover. Commensurability partitions the set of hyperbolic 3-manifolds into equivalence classes, called commensurability classes. Hyperbolic knot complements appear to be rare in commensurability class, in fact Reid and Walsh have conjectured that there are at most three hyperbolic knot complements in a commensurability class. I will discuss recent progress on Reid and Walsh's conjecture with a focus on the open problems in this area.