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 $M^4(L,c)$ 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.

28 pages, 24 figures. Version accepted for publication in Revista Matemática Complutense, with new examples, explanations and figures, according to the referee's suggestions

References in corpus (3)

• G-degree for singular manifolds
• Rigid gems in dimension n
• Classifying compact 4-manifolds via generalized regular genus and G-degree

Cited by in corpus (1)

• Trisections of PL 4-manifolds arising from colored triangulations