Knots are determined by their complements

This answers a question apparently first raised by Tietze T, p. 83. It was previously known that there were at most two knots with a given complement CGLS, Corollary 3. The notion of equivalence of knots can be strengthened by saying that K and K' are isotopic if the above homeomorphism h is isotopic to the identity, or, equivalently, orientation-preserving. The analog of Theorem 1 holds in this setting too: if two knots have complements that are homeomorphic by an orientation-preserving homeomorphism, then they are isotopic. Theorem 1 and its orientation-preserving version are easy consequences of the following theorem concerning Dehn surgery.