From framed links to crystallizations of bounded 4-manifolds

It is well-known that every 3-manifold $M^3$ may be represented by a framed link (L, c), which indicates the Dehn-surgery from $S^3$ to $M^3 = M^3(L, c)$; moreover, $M^3$ is the boundary of a PL 4-manifold $M^4 = M^4(L, c)$, which is obtained from $D^4$ by adding 2-handles along the framed link (L, c). In this paper we study the relationships between the above representations and the representation theory of general PL-manifolds by edge-coloured graphs: in particular, we describe how to construct a 5-coloured graph representing $M^4 = M^4(L, c)$, directly from a planar diagram of (L, c). As a consequence, relations between the combinatorial properties of the link L and both the Heegaard genus of $M^3 = M^3(L, c)$ and the regular genus of $M^4 = M^4(L, c)$ are obtained.