# Ribbonness of a stable-ribbon surface-link, II. General case

## TL;DR

This paper proves that all stable-ribbon surface-links are ribbon surface-links and establishes conditions under which connected sums and certain surface-links are ribbon, generalizing previous results in the field.

## Contribution

It generalizes the understanding of stable-ribbon surface-links, showing their equivalence to ribbon surface-links and characterizing their behavior under connected sums.

## Key findings

- Stable-ribbon surface-links are all ribbon surface-links.
- Connected sum of surface-links is ribbon if and only if both summands are ribbon.
- Characterization of when surface-links with ribbon components are ribbon.

## Abstract

It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences, so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are given. One observation is that a connected sum of two surface-links is a ribbon surface-link if and only if both the connected summands are ribbon surface-links. The other observation is a characterization of when a surface-link consisting of ribbon surface-knot components becomes a ribbon surface-link.