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Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and $K_2$ are necessarily isotopic (obviously they must be if the ribbon disks are stably equivalent)?

For the $n=1$ case Nakanishi and Nakagawa have counterexamples to this in On Ribbon Knots, Math. Sem. Notes Kobe Univ. 5, 1982. However, their examples don't seem generalize in an obvious way to higher dimensional ribbon knots.

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    $\begingroup$ In "A Quick Trip Through Knot Theory" Fox gives examples 10 and 11. I presume these are examples of what you are speaking about. As I recall, the complements are mirror images of each other (maybe inverses). Kaneobu generalized the construction of examples 10,11,12, in the case $n=1$, and showed that all of the generalizations from 12 were twist-spun knots. In addressing the problem, I would try to generalize Kaneobu's technique. $\endgroup$ – Scott Carter Mar 3 '14 at 23:25

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