On delta-unknotting operation (Q1327622)

A \(\Delta\)-unknotting operation is a kind of unknotting operations, a finite sequence of which changes any knot into the unknot. A typical example is a crossing change operation (ordinary unknotting operation). Suppose that a knot \(k'\) is obtained from another knot \(k\) by a \(\Delta\)- unknotting operation. It is shown that there are infinitely many non- homeomorphic \(\Delta\)-unknotting operations which also change \(k'\) into \(k\). Compare that if \(k'\) is a doubled knot and \(k\) is the unknot, the ordinary unknotting operation changing \(k'\) into \(k\) is unique up to homeomorphism [\textit{T. Kobayashi}, Kobe J. Math. 6, 53-62 (1989; Zbl 0688.57007); \textit{M. Scharlemann} and \textit{A. Thompson}, Comment. Math. Helv. 64, No. 4, 527-535 (1989; Zbl 0693.57004)].