Ribbonness on classical link

TL;DR

This paper proves that any link bounding a proper oriented surface in 4-space can be embedded as a ribbon surface, confirming that all slice knots are ribbon knots and answering a longstanding question.

Contribution

It establishes that all links bounding proper surfaces in 4-space can be realized as ribbon surfaces, affirming that every slice knot is a ribbon knot.

Findings

Every slice knot is a ribbon knot.

Links bounding proper surfaces can be embedded as ribbon surfaces.

Provides a positive answer to R. H. Fox's question.

Abstract

It is shown that if a link in 3-space bounds a proper oriented surface (without closed component) in the upper half 4-space, then the link bounds a proper oriented ribbon surface in the upper half 4-space which is a renewal embedding of the original surface. In particular, every slice knot is a ribbon knot, answering an old question by R. H. Fox affirmatively.