Georgia Institute of Technology College of Sciences

Dehn surgery (with a fixed slope p/q) associates, to a knot in S^3, a 3-manifold M with first homology isomorphic to the integers mod p. One might wonder if this function is one-to-one or onto; Cameron Gordon (1978) conjectured that it is never injective nor surjective. The surjectivity case was established a decade later, while the injectivity case was only recently proven by Hayden, Piccirillo, and Wakelin. We will survey this latter result and its proof.