# Cosmetic Surgery Conjecture on S3

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Zhongtao Wu, Princeton University
Fine Hall 314

It has been known over 40 years that every closed orientable 3-manifold is obtained by surgery on a link in $S3$. However, a complete classification has remained elusive due to the lack of uniqueness of this surgery description. In this talk, we discuss the following uniqueness theorem for Dehn surgey on a nontrivial knot in $S3$. Let $K$ be a knot in $S3$, and let $r$ and $r'$ be two distinct rational numbers of same sign, allowing $r$ to be infinite; then there is no orientation preserving homeomophism between the manifolds obtained by performing Dehn surgery of type $r$ and $r'$, respectively. In particular, this result implies the Knot Complement Theorem of Gordon and Luecke.