Heegaard Floer Surgery Formula and Cosmetic Surgeries

Abstract

Two Dehn surgeries on a knot are called cosmetic if they yield homeomorphic three-manifolds. We show for a certain family of null-homologous knots in any closed orientable three-manifold, if the knot admits cosmetic surgeries with a pair of positive surgery coefficients, then the coefficients are both greater than $1$. In addition, for this family of knots, we show that $1/q$ Dehn surgery for $q$ at least $2$ is not homeomorphic to the original three-manifold. The proofs of these results use the mapping cone formula for the Heegaard Floer homology of Dehn surgery in terms of the knot Floer homology of the knot; we provide a new proof of this formula for integer surgeries in $\text{Spin}^c$ structures with nontorsion first Chern class.