In Gompf-Stipsicz book, we're presented with the Akbulut cork, and a brief explanation of why it is contractible (see below).
Would someone be able to explain what homotopy he is referring too? I thought at first he is referring to the fact that any link can be unknotted by pushing to the interior of the 4-manifold, but this homotopy doesn't seem to simplify the underlying handles with it.
Up to homotopy (by deflating the $\times D^2$ of the 2-handle first), attaching a 4-dimensional 2-handle is the same as attaching its core (a $D^2$) along its attaching sphere (a $S^1$).
This means the homotopy class of the attaching map $S^1\to h_0\cup h_1$ uniquely determines the homotopy type of the end manifold. In particular, the framing does not change the homotopy type.
So, as long as we stay away from the 1-handle, we can isotope the attaching sphere (which is the framed circle in the diagram) as much as we want and keep the homotopy type. In particular, we may undo the clasp. This means the end manifold is homotopy equivalent to the one with diagram a link of two knots, one a 1-handle and the other a 0-framed 2-handle, i.e., $B^4$.