Alternative proof of the ribbonness on classical link

TL;DR

This paper provides an alternative proof for a result linking the existence of certain surfaces in 4-space to the ribbonness of classical links, enhancing understanding of their geometric properties.

Contribution

It offers a new proof of a known theorem connecting surfaces in 4-space to ribbon links, simplifying and clarifying the original argument.

Findings

Confirmed the equivalence between bounding a proper surface and bounding a ribbon surface in 4-space.

Provided a more accessible proof of the ribbonness property for classical links.

Strengthened the theoretical foundation for studying link surfaces in 4-dimensional topology.

Abstract

Alternative proof is given for an earlier presented result that if a link in 3-space bounds a compact oriented proper surface (without closed component) in the upper half 4-space, then the link bounds a ribbon surface in the upper half 4-space which is a boundary-relative renewal embedding of the original surface.