Gordon-Luecke theorem

This fact is related to: knot theory

Statement

Let $K_{1}$ and $K_{2}$ be two knots (homeomorphic copies of $S^{1}$) in $S^{3}$. Then, any homeomorphism between the knot complements $S^{3}\setminus K_{1}$ and $S^{3}\setminus K_{2}$ extends to a self-homeomorphism of $S^{3}$. Thus $K_{1}$ and $K_{2}$ sit in the same way (upto possibly a reversal of orientation) inside $S^{3}$.