A graph-theoretical representation of PL-manifolds — A survey on crystallizations

Dotted Links, Heegaard Diagrams, and Colored Graphs for PL 4-manifolds

“…In fact, by construction, ∂Λ(L, c) has a vertex for every boundary vertex ofΛ(L, c), and a j-colored edge (j ∈ Δ 3 ) for every {j, 4}-colored path inΛ(L, c) joining two boundary vertices. Since the boundary graph always represents the boundary manifold (see [11] for details), the second part of Proposition 2.1 follows, too.…”

“…Moreover,Λ(L, c) admits as its boundary graph (see [11] for details) the 4-colored graph Λ(L, c), which represents the …”

“…First, we have to recall some fundamental notions and terminology of crystallization theory; for a much more detailed account, we refer to [11], where a useful bibliography may also be found. An (n + 1)-colored graph is a pair (Γ, γ), where Γ = (V (Γ), E(Γ)) is a multigraph (i.e.…”

“…The aim of the present paper is to establish a connection between the 4-manifold representation method by dotted framed links (or equivalently-in the closed caseby Heegaard diagrams) and the so called crystallization theory, which visualizes general PL-manifolds by means of edge-colored graphs (see [11], [1], [5], [10], [14], [16], [22],. .…”

Regular Genus and Products of Spheres

“…Colored graphs represent a useful combinatorial approach to the topology of piecewise linear (PL) manifolds. For more details see for example the survey papers [1], [5], [7], [10], and [18]. An n-pseudocomplex K(G) can be associated to (G, c) by the following rules.…”