Knotted concordances of slice links

Question

Are there any examples of a link $L$ such that:

1. $L$ is (strongly) slice, meaning that there exists a properly embedded collection $C$ of $n=|L|$ disjoint annuli in $S^3\times [0,1]$ such that $C\cap S^3\times 0=L$ and $C\cap S^3\times 1$ is the $n$-component unlink,
2. $L$ has an unknotted component $U$,
3. For every concordance $C$ from $L$ to the unlink as in point 1, the annulus $A$ containing $U$ is not "trivial", meaning that $(S^3\times [0,1],A)$ is not isomorphic to $(S^3\times [0,1],U\times[0,1])$?

I didn't specify if the conditions above should hold in $TOP$ or $DIFF$ since I think it would be interesting to know if there is an answer in both categories.

In case that there are no known examples, what kind of invariants may be able to detect this phenomenon? More explicitly: if I have a link $L$ that satisfies points 1 and 2, how can I prove that point 3 holds?

Sadly, I couldn't even find reasonable candidates that satisfy 1 and 2 and for which 3 was not easily seen to be false.

Answer 1

I think this is likely an unknown question. Namely, the negation of 3) would follow from 1) and 2) if

• strongly slice links are strongly ribbon (which seems to be open)
• ribbon disks bounding the unknot are unknotted, considered in this paper more generally for fibered slice knots.

Let’s assume these two bullet points have a positive solution. Then given a strongly slice link $L$ as in 1), capping off the unlink would give disjoint disks bounding $L$ in the 4-ball. Then $L$ should bound ribbon disks if the first bullet point is true. The ribbon disk bounding the unknot in assumption 2) would be unknotted if the second bullet point it true. Cut out a ball neighborhood intersecting all of the ribbon disks in standard disks bounding the unlink to get a concordance with the unknotted component annulus standard.

Given that two open questions would imply a negative solution to your question, it seems plausible that the question is open (or at least unlikely that there is a positive answer in the literature, since it would imply a negative solution two one of two open questions that seem to be fairly well-known).