The Knot Complement Conjecture

Let $K$ be a knot in a closed connected orientable $3$--manifold $Y$, with exterior $M=Y\setminus\operatorname{int}N(K)$ irreducible and not homeomorphic to the solid torus. A slope is an isotopy class of unoriented essential simple closed curves on $\partial M$; write $Y_r(K)$ for the result of Dehn surgery along $r$, and let the meridian of $K$ be the slope bounding a disk in $N(K)$. Two slopes are equivalent if a homeomorphism of the knot exterior takes one to the other. Knot complement conjecture. If there is a non-trivial slope $r$ such that there is an orientation-preserving homeomorphism

$$Y_r(K)\cong Y,$$

then $r$ is equivalent to the meridian of $K$. This is the surgery formulation of the claim that knots satisfying the stated hypotheses are determined by their complements. The supplied source notes that knots in $S^3$ are determined by their complements, while giving no resolution of the conjecture in the generality stated.