  Simple crystallizations are edge-coloured graphs representing PL 4-manifolds
with the property that the 1-skeleton of the associated triangulation equals
the 1-skeleton of a 4-simplex. In the present paper, we prove that any
(simply-connected) PL $4$-manifold $M$ admitting a simple crystallization
admits a special handlebody decomposition, too; equivalently, $M$ may be
represented by a framed link yielding $\mathbb S^3$, with exactly $\beta_2(M)$
components ($\beta_2(M)$ being the second Betti number of $M$). As a
consequence, the regular genus of $M$ is proved to be the double of
$\beta_2(M)$. Moreover, the characterization of any such PL $4$-manifold by
$k(M)= 3 \beta_2(M)$, where $k(M)$ is the gem-complexity of $M$ (i.e. the
non-negative number $p-1$, $2p$ being the minimum order of a crystallization of
$M$) implies that both PL invariants gem-complexity and regular genus turn out
to be additive within the class of all PL $4$-manifolds admitting simple
crystallizations (in particular: within the class of all "standard"
simply-connected PL 4-manifolds).
