The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic.
These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a nontrivial knot in the 3-sphere can yield the 3-sphere.
The theorem was proved by Cameron Gordon and John Luecke.
Essential ingredients of the proof are their joint work with Marc Culler and Peter Shalen on the cyclic surgery theorem, combinatorial techniques in the style of Litherland, thin position, and Scharlemann cycles.
Another method is to twist along an annulus spanning two components.