Are two slice surfaces with minimal genus isotopic? For a link $L\subset S^3$ and two Seifert surfaces (edit: a better name would be slice surfaces as the comments below 1 2 point out) with minimal genus $S_1,S_2\subset B^4$, I have the following questions:

*

*Is it true that the surfaces are isotopic?

*And if the knots are algebraic?

*I am interested in the Hopf link, so the surface is a cylinder, is it true if the surfaces have genus zero?

I know of the existence of the paper Livingston - Surfaces bounding the unlink and Using a 4th dimension to make Seifert surfaces isotopic?
but it answers a particular case and I hope that there are updates since 1980 (does not seem so from the citations).
 A: The question is very loaded and the question would almost require a survey...
Anyway, the answer to your questions is mostly no. Let $S \subset S^4$ be a 2-knot (i.e. an embedded 2-sphere), $p \in S$ a point, and remove a small ball around $p$. The complement of $S$ in the complement of the ball is a slice disc $D$ for the unknot, and the fundamental group of the complement of $D$ is isomorphic to the complement of the fundamental group of $S^4\setminus S$, which can be non-trivial. (It is possible that $D$ is trivial if and only if $S$ is, but I don't want to make the claim without having properly checked). So, any $2$-knot with non-cyclic fundamental group gives you a slice disc for the unknot which is not isotopic to the standard one.
You can plant the construction into any minimal-genus Seifert surface, and probably you get the same construction for any knot. (Perhaps using the Alexander polynomial is a good way to show that you get different things, without worrying too much about Seifert--van Kampen.)
Now, you might ask: what about slice surfaces whose complement has cyclic fundamental group? Then the answer depends on the category. For locally-flat slice discs, Freedman showed that they are topologically isotopic. There's a nice preprint of Conway, Piccirillo, and Powell where they treat higher-genus surfaces and they give a complete classification in terms of algebro-topological data (Blanchfield forms). In the smooth category, things get pretty weird: Kyle Hayden has examples of slice discs that are topologically isotopic but not smoothly isotopic (see here) and of pushed-in Seifert surfaces that stay non-isotopic in $B^4$ (note that these automatically have cyclic fundamental group).
Back on the subject of pushed-in Seifert surfaces, Hayden, Kim, M. Miller, Park, and Sundberg proved that they can be topologically non-isotopic and topologically isotopic but not smoothly isotopic (see here and the Quanta Magazine article on their result).
