Abstract: A pair of distinct slopes for a knot K is called a $\mbox{\it cosmetic surgery pair}$ if the Dehn surgeries along those slopes yield the same oriented 3-manifold. Gordon conjectured that such pairs do not exist.
I will describe a theorem that says any potential cosmetic surgery pairs on a hyperbolic knot K belong to a finite list of slopes, whose size is determined by the systole (shortest closed geodesic) in the complement of K. For a typical knot, this list has no more than 10 pairs of slopes. This makes it feasible to check the remaining pairs by computer and prove that K has no cosmetic surgeries at all. For instance, prime knots up to 15 crossings have no cosmetic surgeries. This is joint work with Jessica Purcell and Saul Schleimer.
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