Faithful equivalence of equivalent ribbon surface-links

Abstract

A chord graph in [Formula: see text]-space is constructed from a ribbon surface-link in [Formula: see text]-space. In earlier papers, the three moves on the diagrams of chord graphs (namely, the chord diagrams) were introduced to describe the faithful equivalence of a ribbon surface-link. In this paper, it is shown that any two equivalent ribbon surface-links are faithfully equivalent, so that any chord diagrams of any two equivalent ribbon surface-links are connected by a finite number of these three moves. By combining it with an earlier result, it is shown that any two TOP-equivalent ribbon surface-links are equivalent. In other words, there is no exotic ribbon surface-link, generalizing an earlier result on the trivial ribbon surface-knot. In another earlier result, the three moves on the chord diagrams were modified into the 16 moves on the chord diagrams without base crossing. In this paper, further modified moves of the 16 moves on the chord diagrams without base crossing are also introduced to describe how the set of ribbon torus-links is produced from the set of welded virtual links.