Kirby diagrams and 5-colored graphs representing compact 4-manifolds

It is well-known that in dimension 4 any framed link $(L,c)$ uniquely represents the PL 4-manifold $M^4(L,c)$ obtained from ${\mathbb{D}}^4$ by adding 2-handles along $(L,c)$. Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram $(L^{(\ast )},d)$), the associated PL 4-manifold $M^4(L^{(\ast )},d)$ is obtained from ${\mathbb{D}}^4$ by adding 1-handles along the dotted components and 2-handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edge-colored graphs: in particular, we describe how to construct algorithmically a (regular) 5-colored graph representing $M^4(L^{(\ast )},d)$, directly “drawn over” a planar diagram of $(L^{(\ast )},d)$, or equivalently how to algorithmically obtain a triangulation of $M^4(L^{(\ast )},d)$. As a consequence, the procedure yields triangulations for any closed (simply-connected) PL 4-manifold admitting handle decompositions without 3-handles. Furthermore, upper bounds for both the invariants gem-complexity and regular genus of $M^4(L^{(\ast )},d)$ are obtained, in terms of the combinatorial properties of the Kirby diagram.