Cosmetic surgeries on knots in $S^3$

TL;DR

This paper investigates conditions under which two Dehn surgeries on a knot in the 3-sphere produce homeomorphic manifolds, proving that purely cosmetic surgeries must have opposite slopes using Heegaard Floer homology techniques.

Contribution

It establishes that purely cosmetic Dehn surgeries on knots in $S^3$ can only occur with opposite slopes, utilizing a new Dehn surgery formula for correction terms in Heegaard Floer homology.

Findings

Purely cosmetic surgeries require opposite slopes.

Dehn surgery formula for correction terms developed.

Supports conjecture on cosmetic surgeries in $S^3$.

Abstract

Two Dehn surgeries on a knot are called {\it purely cosmetic}, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in $S^3$, we show that the two surgery slopes must be the opposite of each other. One ingredient of our proof is a Dehn surgery formula for correction terms in Heegaard Floer homology.