Knots with unknotting number one are determined by their complements (Q1123447)

A knot K in \(S^ 3\) is said to be determined by its complement if a homeomorphism of the complements \(S^ 3-K\) and \(S^ 3-K'\) yields the equivalence of K and \(K'\). In this paper, the author proves the theorem indicated by the title as an immediate corollary of the main theorem: Theorem 1. If a knot K is a banded Hopt link, then K is determined by its complement. A proof depends on the recent results obtained by Culler- Gordon-Luecke-Shalen and D. Gabai. It should be noted that C. McA. Gordon and J. Luecke recently proved that any knot is determined by its complement.