On tunnel number one links with surgeries yielding the 3-sphere
\textit{C.McA. Gordon} and \textit{J. Luecke} [J. Am. Math. Soc. 2, No.~2, 371--415 (1989; Zbl 0678.57005)] showed that knots are determined by their complements. That is, any non-trivial Dehn surgery on a non-trivial knot in the \(3\)-sphere does not yield the \(3\)-sphere. But there is a non-trivial link in the \(3\)-sphere which admits a non-trivial Dehn surgery yielding the \(3\)-sphere. A trivial example is a link having a trivial component or having a non-separating essential annulus in its exterior. Nontrivial examples of such links have been constructed. In particular, \textit{J. Berge} [Embedding the exteriors of one-tunnel knots and links in the \(3\)-sphere, unpublished manuscript] gave such examples of two component links which were tunnel number one. In the present paper, the author extends Berge's examples to construct infinitely many such tunnel number one links in the \(3\)-sphere.
scientific article; zbMATH DE number 5692890
scientific article