# The cosmetic surgery conjecture and Heegaard Floer homology

# 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). Heegaard Floer homology has proved to be a powerful tool for approaching this problem, leading to several partial results; I will show that these results can be improved to further obstruct cosmetic surgeries. Specifically, if a knot in S^3 admits purely cosmetic surgeries, then the surgery slopes must be +/- 2 or +/- 1/q. Moreover, for any given knot there is an upper bound for q in terms of the Heegaard Floer thickness, so there are at most finitely pairs of slopes which are not ruled out. Using a computer we check that the conjecture holds for all prime knots with up to 15 crossings, as well as arbitrary connected sums of such knots. The genus of K is also bounded by a function of the thickness; a simple consequence is that the conjecture holds for any (quasi)alternating knot with genus not equal to 2.