Note on surface-link of trivial components

TL;DR

This paper extends the understanding of ribbon surface-links by showing that trivial components in certain disconnected surfaces are ribbon links, and demonstrates the existence of non-ribbon links with isomorphic fundamental groups.

Contribution

It generalizes previous results to disconnected surfaces with multiple non-sphere components and constructs examples of trivial-component links with different ribbon properties.

Findings

Trivial components in certain disconnected surfaces form ribbon surface-links.

Existence of trivial-component links with isomorphic fundamental groups but different ribbon properties.

Extension of ribbon link characterization to more complex surface configurations.

Abstract

As a previous result, it has shown that every sphere-link consisting of trivial components is a ribbon sphere-link. In this note, it is shown that for every closed oriented disconnected surface F with just one non-sphere component, every F-link consisting of trivial components is a ribbon surface-link. Further, it is shown that for every closed oriented disconnected surface F containing at least two non-sphere components, there exist a pair of a ribbon F-link and a non-ribbon F-link that consist of trivial components and have meridian-preservingly isomorphic fundamental groups.