5 problems

Cosmetic surgery conjecture. The conjecture predicts that non-trivial knots admit no truly cosmetic surgeries.

Let $M$ be a $1$-cusped hyperbolic $3$-manifold, and let $\text{}alpha$ and $\text{}beta$ be inequivalent slopes on $M$. They are truly cosmetic if $M(\alpha)$ and $M(\beta)$ admit…

Contact cosmetic surgery conjecture. Any Legendrian knot in $(Y,si)$ that is not smoothly an unknot admits no cosmetic contact surgeries.

Let $K$ be a hyperbolic knot in the 3-sphere $S^3$. A pair of Dehn surgeries on $K$ is chirally cosmetic if the resulting 3-manifolds are orientation-reversingly homeomorphic, and…

Let $K$ be a knot in the 3-sphere $S^3$. A pair of Dehn surgeries on $K$ is chirally cosmetic if the resulting 3-manifolds are orientation-reversingly homeomorphic; it is integral…