Purely Cosmetic Surgery Conjecture

Let $K$ be a knot, and let $S^3_r(K)$ denote the manifold obtained by surgery along $K$ with slope $r$. Purely Cosmetic Surgery Conjecture. If

$$S^3_r(K) \cong S^3_{r'}(K)$$

via an orientation-preserving homeomorphism, then either $K$ is the unknot or $r=r'$. The paper proves that two infinite families of knots satisfy this conjecture using the Jones polynomial and related surgery obstructions, while the conjecture remains open in general.