show/hide this revision's text 2 take Greg's comment into account

The slice-ribbon conjecture asserts that all slice knots are ribbon.

This assumes the context:

1) A `knot' is a smooth embedding $S^1 \to S^3$. We're thinking of the 3-sphere as the boundary of the 4-ball $S^3 = \partial D^4$.

2) A knot being slice means that it's the boundary of a 2-disc smoothly embedded in $D^4$.

3) A slice disc being ribbon is a more fussy definition -- a slice disc is in ribbon position if it can be isotoped (keeping its boundary fixed) so that the distance function $d(p) = |p|^2$ is Morse on the slice disc , having only saddle and local-min type critical points, with the saddle-type critical points corresponding to larger critical values that the having no local minima ie $d(p_{\text{saddle}}) > d(q_{\text{loc.min.}})$maxima. A slice knot is a ribbon knot if it's slice and one of its slice discs is has a ribbon discposition.

My question is this. All the above definitions have natural generalizations to links in $S^3$. You can talk about a link being slice if it's the boundary of disjointly embedded discs in $D^4$. Similarly, the above ribbon definition makes sense for slice links. Are there simple examples of $n$-component links with $n \geq 2$ that are slice but not ribbon? Presumably this question has been investigated in the literature, but I haven't come across it. Standard references like Kawauchi don't mention this problem (as far as I can tell).

show/hide this revision's text 1

slice-ribbon for links (surely it's wrong)

The slice-ribbon conjecture asserts that all slice knots are ribbon.

This assumes the context:

1) A `knot' is a smooth embedding $S^1 \to S^3$. We're thinking of the 3-sphere as the boundary of the 4-ball $S^3 = \partial D^4$.

2) A knot being slice means that it's the boundary of a 2-disc smoothly embedded in $D^4$.

3) A slice disc being ribbon is a more fussy definition -- a slice disc is ribbon if it can be isotoped (keeping its boundary fixed) so that the distance function $d(p) = |p|^2$ is Morse on the slice disc, having only saddle and local-min type critical points, with the saddle-type critical points corresponding to larger critical values that the local minima ie $d(p_{\text{saddle}}) > d(q_{\text{loc.min.}})$. A knot is a ribbon knot if it's slice and one of its slice discs is a ribbon disc.

My question is this. All the above definitions have natural generalizations to links in $S^3$. You can talk about a link being slice if it's the boundary of disjointly embedded discs in $D^4$. Similarly, the above ribbon definition makes sense for slice links. Are there simple examples of $n$-component links with $n \geq 2$ that are slice but not ribbon? Presumably this question has been investigated in the literature, but I haven't come across it. Standard references like Kawauchi don't mention this problem (as far as I can tell).