Who invented special framed links as a way to specify four-manifolds? It is possible to specify (smooth, oriented, compact) four-manifolds with special framed links:

$$I \times \mathbb{R}P^3$$
Such a link is divided in two sublinks, one ordinary link representing the 2-handles (and their attaching curves) and an unlinked, unknotted link representing the 1-handles.
This notation is used e.g. in Eugénia César de Sá, A link calculus for 4-manifolds (1979), and in more modern works like the book by Gompf &amp; Stipsicz (chapter 5.4), where it is attributed to Selman Akbulut (in works from 1977 to 1979), who in turn thanks Robion Kirby and cites his influential A calculus for framed links in $S^3$ (back then yet to appear), which doesn't talk about 1-handles in 4-manifolds though.
What intrigues me is that de Sá and Akbulut don't cite each other, so I'm somehow suspecting that maybe the notion was developed by someone else (Kirby? Rourke?) and de Sá and Akbulut used it without particularly citing it because everyone knew where it came from?.
Or is the situation yet different and Akbulut &quot;only&quot; developed special framed links and de Sá developed &quot;only&quot; the $\Gamma$-moves, which tell are the equivalence transformations of special framed links?
Here is my answer, hope it clarifies the question.

A Link Calculus for 4 Manifolds was taken from part of my PhD Thesis in Warwick, U.K, with the title Automorphisms of 3 Manifolds and Representations of 4 Manifolds, 1977, which, obviously, was written before. 

This article is from the proceedings  of a Conference in Low Dimensional Topology that took place in Sussex 1977.

I knew the Kirby Calculus for 3 Manifolds, both from preprints that were circulating and from Kirby because I attended several seminars given by him.
I also knew the representation of a 1 handle as as circle with a dot from the same source. And that's all.

Kirby is referred in both my thesis and in this article.
There was a lot of "common knowledge" for orientable manifolds at the time.

All the other is original from my thesis (but I was only an unknown PhD student...)

The name Gamma moves that appear in the framed links for non-orientable manifolds and you refer to, comes from my name. A short name for Eugénia  is, in my country, Gena, so I put Gamma because it's the greek letter for G.

Eugénia  César de Sá
Can try to scan the 4 Manifold part of my thesis ( have to find the originals ☺) and send you by mail, if you want to. Or photopcopy them and post them to you