I would like to see why the following two statements in Kirby's list of problem are equivalent:
If $K_1$ and $K_2$ are knots in a closed oriented 3-manifold $M$ whose complements are homeomorphic via orientation-preserving homeomorphism then there exists an orientation-preserving homeomorphism of $M$ taking $K_1$ to $K_2$.
Let $M$ be an oriented 3-manifold with torus boundary. If $M(r_1)\cong M(r_2)$ for inequivalent slopes, then the homeomorphism is orientation-reversing.
(Here $M(r_1)$, $M(r_2)$ are the result of the Dehn filling of $M$ with slope $r_1$, $r_2$.)