A homotopy 4-sphere that is built without 1-handles can be encoded as a $n$-component link with an integral
Dehn surgery to $\#^n(S^1\times S^2)$. I'll describe a program to prove that such spheres are smoothly standard
in the case that $n=2$ and one component of the link is fibered, which has been carried out in joint work with Alex Zupan in the case that the fibered knot is a generalized square knot. I'll discuss how this relates to the problem of classifying ribbon disks for a fibered knot, and, time permitting, I'll outline how the theory of trisections connects this work to the Andrews-Curtis Conjecture and the Generalized Property R Conjecture.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/TopologySeminar