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


For the definition of ribbon link see: [ribbon links - counterexamples](https://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:


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



Boundary links and ribbon links.
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.](http://arxiv.org/abs/0709.2968)


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


