Morse function on slicing disk complement determines ribbon?

Question

It is well-known that given a ribbon knot and the corresponding slicing disk in the 4-ball, the distance function (maybe squared) to the origin defines a Morse function in the complement of the slicing disk with critical points of index <=2. My question is that if we are given a (smoothly) slicing disk whose complement admits a Morse function with critical points of index <=2, but not necessarily the distance function, then do we know the knot is ribbon or are there (potential) counterexamples?

Answer 1

As far as I know and understand, this is an open question. If $D\subset D^4$ is a slicing disc for a knot $K = \partial D \subset S^3$, then the following implications hold:

$D$ is ribbon $\Longrightarrow$ $D^4\setminus N(D)$ has a handle decomposition with 0, 1, and 2-handles only (no 3-handles) $\Longrightarrow$ the induced map $\pi_1(S^3\setminus K) \to \pi_1(D^4\setminus D)$ is surjective.

Here $N(D)$ is an open tubular neighborhood for $D$.

The second condition is equivalent to the existence of a Morse function as you required (if I understand it correctly).

The third is the "homotopically ribbon" condition that was introduced by Casson and Gordon.

It seems to me that it is unknown whether any of the two implications can be reversed, but I haven't seen it written explicitly anywhere.