Unknotting and fusion numbers of ribbon 2-knots
The unknotting number \(u(K)\) of a 2-knot \(K\) is the least number of added 1-handles, in an appropriate manner, to \(K\) giving an unknotted surface. Since a ribbon 2-knot is obtained from \(n+1\) standardly embedded (i.e. unlinked, unknotted in the sense that they bound disjoint balls) 2-spheres by adding \(n\) 1-handles for some \(n\), the fusion (or ribbon) number \(f(K)\) of a ribbon 2-knot \(K\) is the least possible \(n\) for \(K\). As generally holds \(u(K)\leq f(K)\), the authors are searching for cases when the strict inequality holds. They prove: For any integers \(m\) and \(n\) with \(0<m\leq n\), there exists a ribbon 2-knot \(K\) such that \(u(K)=m\) and \(f(K)=n\). Since the spun 2-knot \(K^*\) of a 1-knot \(K\) is a ribbon 2-knot the other offered result is: For a nontrivial spun torus knot \(T^*_{p,q}\) with \(1<p<q\) it holds that \(u(T^*_{p,q})=1\), while \(f(T^*_{p,q}) =p-1\).
scientific article; zbMATH DE number 1117469