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

TL;DR

This paper develops an algorithmic method to construct 5-colored graphs and triangulations of compact 4-manifolds from Kirby diagrams, enabling analysis of their topological invariants and handle decompositions.

Contribution

It introduces a new algorithmic approach to represent 4-manifolds via edge-colored graphs directly from Kirby diagrams, linking diagrammatic and combinatorial topology.

Findings

Constructs 5-colored graphs from Kirby diagrams

Provides bounds for gem-complexity and regular genus

Enables triangulation of certain 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 $(L^{(*)},d)$), the associated PL 4-manifold $M^4(L^{(*)},d)$ 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 $M^4(L^{(*)},d)$, directly "drawn over" a planar diagram of $(L^{(*)},d)$, or equivalently how to algorithmically obtain a triangulation of $M^4(L^{(*)},d)$. 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 $M^4(L^{(*)},d)$ are obtained, in terms of the combinatorial properties of the Kirby diagram.