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

Abstract

It is well-known that in dimension 4 any framed link (L, c) uniquely represents the PL 4-manifold $$M^4(L,c)$$ M 4 ( L , c ) obtained from $${\mathbb {D}}^4$$ 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$$(L^{(*)},d)$$ ( L ( ∗ ) , d ) ), the associated PL 4-manifold $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) is obtained from $${\mathbb {D}}^4$$ 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 $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) , directly “drawn over” a planar diagram of $$(L^{(*)},d)$$ ( L ( ∗ ) , d ) , or equivalently how to algorithmically obtain a triangulation of $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , 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 $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) are obtained, in terms of the combinatorial properties of the Kirby diagram.