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Jonathan Hanselman : The cosmetic surgery conjecture and Heegaard Floer homology

The cosmetic surgery conjecture states that no two surgeries on a given knot produce the same 3-manifold (up to orientation preserving diffeomorphism). Floer homology has proved to be a powerful tool for approaching this problem; I will survey partial results that are known and then show that these results can be improved significantly. If a knot in S^3 admits purely cosmetic surgeries, then the surgery slopes are +/- 2 or +/- 1/q, and for any given knot we can give an upper bound for q in terms of the Heegaard Floer thickness. In particular, for any knot there are at most finitely many potential pairs of cosmetic surgery slopes. With the aid of computer computation we show that the conjecture holds for all knots with at most 15 crossings.

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