Kirby diagrams and 5-colored graphs representing compact 4-manifolds
Maria Rita Casali, Paola Cristofori

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 uniquely represents the PL 4-manifold obtained from by adding 2-handles along . Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram ), the associated PL 4-manifold is obtained from 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 , directly "drawn over" a planar diagram of , or equivalently how to algorithmically obtain a triangulation of . As a consequence, the procedure…
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