0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots

Question

A knot in $S^3$ is called a smoothly slice knot if it bounds a smoothly embedded 2-disk in $D^4$. Every ribbon knot is known to be a smoothly slice knot, and there are known some nontrivial smoothly slice knots: https://en.wikipedia.org/wiki/Slice_knot#Examples.

The Stevedore knot (https://en.wikipedia.org/wiki/Stevedore_knot_(mathematics) is an example of a nontrivial knot which is smoothly slice, and the figure below shows that we can obtain the zero-framed Stevedore knot by blowing down a certain link of unknots.

My question is the following: Similarly, is there another example of a 0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots?

Edit: Can we obtain such a link with the following additional assumptions?

1. the unknots in the link are negatively-weighted (I think this will hold automatically),

2. the link has no isolated unknots,

3. and any two of the unknots of the link are either unlinked or form a Hopf link as in the figure (maybe this is too much to hope).

Answer 1

Yes, you can always do this. In fact, I think you can do it for any framed link, regardless of the framings, the number of components, and the slice assumption. (For simplicity, I will write the answer for the knot case, though.)

Take any diagram for $K$ and any unknotting sequence. This unknotting sequence can be represented by a sequence of blow-ups (with either sign). This gives a diagram in which every component is unknotted, but the framings are off and every two components either link geometrically twice or zero times.

To fix that, we blow up again, undoing a clasp with a blow-up (corresponding to a -1-framed unknotted component). This operation does two things at the same time: it unclasps two components (thus reducing their geometric linking number from 2 to 1, or from 1 to 0), and it lowers the framing on the two components it unclasps. In doing so, the new component we add only links geometrically once with each of the two components it unclasps.

To fix the framings, we can repeat this blow-up operation as much as we want to make all framings negative.

Let me illustrate the procedure with +4-surgery on the mirror of T(2,5). (The pictures are done with KLO, the Kirby calculator.)

We first blow up to unknot.

We now make every pair of knots unlinked or Hopf-linked.

We blow up to make the red and green unknots negatively framed.

We isotope the figure so that we can see clasps that we can unclasp with a $-1$-blow-up.

We blow up at these clasps and iterate the procedure.

We now have a link that respects all of your constraints.

I don't know if you can make the central crossing disappear somehow, but this was not one of the constraints you had, so I haven't thought too long about it.