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.

knowto 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 $S^4$. $\endgroup$ – Ryan Budney Feb 10 '10 at 6:53