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For a real line configuration L, we construct a ribbon surface-link which has the same group as L. If L is a general position line configuration or a central line configuration, the constructed ribbon surface-link has the minimum genus.

A ribbon chord diagram, or simply a chord diagram, of a ribbon surface-link in the 4-space is introduced. Links, virtual links and welded virtual links can be described naturally by chord diagrams with the corresponding moves, respectively. Some moves on chord diagrams are introduced by overseeing these special moves. Then the faithful equivalence on ribbon surface-links is stated in terms of the moves on chord diagrams. This answers questions by Nakanishi and Marumoto affirmatively. The faithful TOP-equivalence on ribbon surface-links derives the same result. By combining a previous result on TOP-triviality of a surface-knot, a ribbon surface-knot is DIFF-trivial if and only if the fundamental group is an infinite cyclic group. This corrects an erroneous proof in Yanagawa's old paper.

A revised proof is given to an assertion on chord diagrams of a ribbon surface-link.

A chord graph in $3$-space is constructed from a ribbon surface-link in $4$-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.

The notion of ribbon-clasp surface-links is a generalization of ribbon surface-links. It is known that any ribbon $T^2$-knot is presented by a welded knot, and then its knot group is isomorphic to the group of the welded knot. In this paper, we define the group of a semi-welded knot. We show that any ribbon-clasp $T^2$-knot is presented by a semi-welded knot and that the knot group of the ribbon-clasp $T^2$-knot is isomorphic to the group of the semi-welded knot.