An unknotting operation on ribbon 2-knots (Q2706859)

The author defines a local move on a ribbon 2-knot diagram, which he calls an HC-move (for ``handle change''). He shows that an HC-move is an unknotting operation, that is, one can transform every ribbon 2-knot into a trivial 2-knot by a finite sequence of HC-moves. Using this fact, he defines the HC-unknotting number for a ribbon 2-knot, and a distance between two ribbon 2-knots, which he calls the HC-Gordian distance. For the normalized Alexander polynomial, \(\Delta(K;t)\), of a ribbon 2-knot \(K\), he defines by \((1/n!)d^n/dt^n\Delta(K;1)\) the \(\alpha_n\)-invariant of \(K\), denoted by \(\alpha_n(K)\). He proves that if \(K\) and \(K'\) are ribbon 2-knots such that \(K'\) is obtained from \(K\) by a single HC-move then \(|\alpha_2(K)-\alpha_2(K')|\leq 1\). Then he applies this result to the calculation of the HC-unknotting numbers of ribbon 2-knots. He also discusses a relation between HC-unknotting number and a 1-handle unknotting number.