Recently been trying to understand the proofs of Gompf and Akbulut that certain 4-manifolds are $S^4$ (these 2 papers: Gompfs paper in Topology Vol. 30 Issue. 1, Akbulut). In which they use a clever 2 and 3 handle cancelling pair to reduce the Kirby diagrams associated to these manifolds.
The part of the proof that I don't quite understand is the way the 2 handle is added.
In an unpublished book on Akbulut's web page Book.pdf he states (on p.14/15),
"Notice that since 3-handles are attached uniquely, introducing a canceling 2- and 3- handle pair is much simpler operation. We just draw the 2-handle as 0-framed unknot, which is $S^2 × B^2$, and then declare that there is a canceling 3-handle on top of it. In a handle picture of 4-manifold, no other handles should go through this 0-framed unknot to be able to cancel it with a 3-handle."
A similar definition for cancelling a 2-3 handle pair is given on p.146 of 4 Manifolds and Kirby Calculus by Gompf and Stipsicz.
My problem is that in both the papers above they add in a 2-3 handle pair but the 2 handle is drawn in such a way that is goes through a 1 handle, which seems to contradict the above cancelling criteria. If anyone could explain this to me it would be greatly appreciated.