Knots with unknotting number one are determined by their complements. (English) Zbl 0677.57006

Thompson, Abigail

Topology 28, No. 2, 225-230 (1989).

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 Hopf 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.

Reviewer: Kunio Murasugi (Toronto)

MSC:

57K10

Knot theory

Keywords:

unknotting number; knots determined by their complements; knot surgery; banded Hopf link