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

Abstract

It is well-known that any framed link (L, c) uniquely represents the 3-manifold M 3 (L, c) obtained from S 3 by Dehn surgery along (L, c), as well as the PL 4-manifold M 4 (L, c) obtained from D 4 by adding 2-handles along (L, c), whose boundary coincides with M 3 (L, c). In this paper we study the relationships between the above representation tool in dimension 3 and 4, and the representation theory of compact PL manifolds of arbitrary dimension by edge-coloured graphs: in particular, we describe how to construct a (regular) 5-colored graph representing M 4 (L, c), directly "drawn over" a planar diagram of (L, c). As a consequence, the combinatorial properties of the framed link (L, c) yield upper bounds for both the invariants gem-complexity and (generalized) regular genus of M 4 (L, c).