Two distinct Knots cannot have the same knot complement (Gordon and Luecke 1989).
References
Cipra, B. ``To Have and Have Knot: When are Two Knots Alike?'' Science 241, 1291-1292, 1988.
Gordon, C. and Luecke, J. ``Knots are Determined by their Complements.'' J. Amer. Math. Soc. 2, 371-415, 1989.