Motions of trivial links, and ribbon knots (Q1913303)

First the authors prove a result on the motions of a trivial link of two components in \(S^{n+2}\), the codimension being 2. Then they use this to define an invariant of the 1-fusion ribbon presentations of a ribbon \(n\)-knot, and to give examples which show that for any positive integers \(m>1\) and \(n\), there exists a ribbon \(n\)-knot with at least \(m\) inequivalent 1-fusion ribbon presentations. Finally an \(n\)-dimensional handcuff is defined to be an embedding of two \(n\)-spheres joined by an arc in \(S^{n+2}\). The invariant referred to above is defined in the same way for the case where the two \(n\)-spheres form a trivial link, and it is shown that when two such handcuffs have the same invariant, one can be deformed into the other by a homotopy of the arc in an orientation-preserving homeomorphism of \(S^{n+ 2}\).