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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 the distance function $d(p) = |p|^2$ is Morse on the slice disc and having no local maxima. A slice knot is a ribbon knot if one of its slice discs has a ribbon position.

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).

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    $\begingroup$ Can't d(saddle) > d(min) be attained trivially by extending fingers of the disk toward the center of the ball? If so, then the essence of ribbonness of a slice knot is just eliminating the local maxima on the disk. $\endgroup$
    – Greg Kuperberg
    Commented Jan 14, 2010 at 1:31
  • $\begingroup$ Ah, right. I'll clean up that statement. $\endgroup$
    – Ryan Budney
    Commented Jan 14, 2010 at 1:37

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Ryan, I think this is an open problem. The best related result I know is a theorem of Casson and Gordon [A loop theorem for duality spaces and fibred ribbon knots. Invent. Math. 74 (1983)] saying that for a fibred knot that bounds a homotopically ribbon disk in the 4-ball, the slice complement is also fibred.

More precisely, they are assuming that the knot K bounds a disk R in the 4-ball such that the inclusion

$S^3 \smallsetminus K \hookrightarrow D^4 \smallsetminus R$

induces an epimorphism on fundamental groups. If one glues R to a fibre of the fibration $S^3 \smallsetminus K \to S^1$ to obtain a closed surface F, then the statement is that the monodromy extends from F to a solid handlebody which is a fibre of a fibration $D^4 \smallsetminus R \to S^1$ extending the given one on the boundary.

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As far as I know, extension of slice and ribbon to links is not unique. There are "strong slice", "weak slice", "strong ribbon" and "weak ribbon" for links.

"CHARACTERIZATION OF SLICES AND RIBBONS" (by H.FOX) mentioned these concepts.

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    $\begingroup$ I'm referring to the specific generalization above, but any results positive or negative on any generalization would be nice, I suppose. It doesn't appear that he has a result of this form though. $\endgroup$
    – Ryan Budney
    Commented Jan 19, 2010 at 21:13

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