Quasi-ribbon but non-ribbon surface-link of trivial components

Abstract

An operation on a surface-link preserving the surface-knot components called a quasi-equivalence is introduced, which induces an equivalence relation on the surface-links. A quasi-ribbon surface-link is a surface-link quasi-equivalent to a ribbon surface-link. It is shown that a surface-link of trivial components is a quasi-ribbon surface link if it has only at most one non-spherical component, whereas there exist examples that are not quasi-ribbon surface-links of trivial components when it has at least two aspherical components. For every closed disconnected, orientable surface, a quasi-ribbon but non-ribbon surface-link of trivial components is constructed. This construction is done by applying a positive solution to Cochran's conjecture, meaning that the sphere-knot obtained from the anti-parallel sphere-link of every non-ribbon sphere-knot by surgery along an s-nontrivial or s-trivial fusion 1-handle is a non-ribbon or trivial sphere-knot, respectively.