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I have two questions about the slice=ribbon conjecture.

(1) If a knot $K \hookrightarrow S^3$ has smooth slice genus $g$, you can ask if it bounds a smooth genus $g$ surface in $S^3 \times [0, -\infty)$, with the function defined by restriction to $[0, -\infty)$ being Morse on the surface without index=0 critical points (maximal points). When $g=0$ this is just asking if the slice knot $K$ has a ribbon disc. I was wondering if there are any knots known with $g \geq 1$ for which such a surface cannot exist. If there are none such known, is there a topological reason why the truth of the slice=ribbon conjecture would also imply the existence of such surfaces?

(2) Are there any potential counterexamples to slice=ribbon (in the same way that there are potential counterexamples to smooth 4-d Poincare [until Akbulut kills them])?

Thanks, Andrew.

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I added a few more tags. Regarding (2) hopefully someone like Ruberman will enter the conversation. I don't know any interesting ways of generating knots that I know to be slice but for which I have reason to suspect they maybe aren't ribbon. IMO the smooth Poincare conjecture is in the same situation. We appear to have a deficit of good ways to identify the standard smooth $S^4$. –  Ryan Budney Feb 10 '10 at 6:53
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I believe (based on a conversation with Sylvain Cappell) that the answer to (2) is no. –  Daniel Moskovich Feb 10 '10 at 14:45
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2 Answers 2

There is a paper by Gompf and Scharlemann: Fibered knots and Property 2R, II, which gives an infinite family of two component links which are smoothly slice but not obviously ribbon.

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Looks like they combined two papers together. –  Jim Conant Dec 2 '10 at 16:07
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This paper by Gompf exhibits a potential counter-example. Has it been established that the candidate given by Gompf is not slice?

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