Question

This question is about the relation between the notions of boundary link and ribbon link.

For the definition of ribbon link see: http://mathoverflow.net/questions/42293/ribbon-links-counterexamples.

An n-component link $L=L_1\cup\dots\cup L_n$ is said to be a boundary link if there exists an orientable surface consisting of n disjoint components $S=S_1\cup\dots\cup S_n$ such that for each $i\leq n$ we have $\partial S_i=L_i$.

There exists boundary links which are not ribbon and ribbon links which are not boundary (see Rolfsen book chapter 5 pag. 140)

It can be shown that every pure ribbon link is a boundary link. (A ribbon link is said to be pure if every ribbon singularity involves only one disc.)

Now my problem is: when is a boundary link ribbon? Clearly every component of such a link must be a ribbon knot so my question is:

• $\textbf{Is every boundary link with ribbon components a ribbon link?}$

Assume the answer to the previous question is yes. Then the following question makes sense:

• $\textbf{Is it true that a link is purely ribbon iff it is a boundary link}$ $\textbf{ with ribbon components?}$

Answer 1

The answer to your first question is no. There are non-ribbon boundary links whose components are unknotted! Indeed the Bing double of a knot is a boundary link with unknotted components, but it has recently been proven by several authors that the Bing double of the figure-8 knot is not slice (hence not ribbon.) See for example this paper.

Edit: The Bing double of the trefoil is also not slice, which is a less subtle calculation using the signature.