Let @math be a simplicial complex with a piecewise linear function @math . The Reeb graph @math is the quotient of @math , where we collapse each connected component of @math to a single point. Let the nodes of @math be all homologically critical points where any homology of the corresponding component of the level set @math changes. Then we can label every arc of @math with the Betti numbers @math of the corresponding @math -dimensional component of a level set. The homology labels give more information about the original complex @math than the classical Reeb graph. We describe a canonical embedding of a Reeb graph into a multi-page book (a star cross a line) and give a unique linear code of this book embedding.
@cite_0: A finitely presented semigroup RSGn is constructed for n ≥ 2. The centre of RSGn encodes uniquely up to rigid ambient isotopy in 3-space all nonoriented spatial graphs with vertices of degree ≤ n. This encoding is obtained by using three-page embeddings of graphs into the three-page book T × I, where T is the cone on three points, and I is the unit segment. The notion of the three-page complexity for spatial graphs is introduced via three-page embeddings. This complexity satisfies the properties of finiteness and additivity under natural operations.
@cite_5: The notion of a basic embedding appeared in research motivated by Kolmogorov–Arnold's solution of Hilbert's 13th problem. Let K,X,Y be topological spaces. An embedding K⊂X×Y is called basic if for every continuous function f:K→R there exist continuous functions g:X→R, h:Y→R such that f(x,y)=g(x)+h(y) for any point (x,y)∈K. Let Ti be an i-od. Theorem. There exists only a finite number of prohibited' subgraphs for basic embeddings into R×Tn. Consequently, for a finite graph K there is an algorithm for checking whether K is basically embeddable into R×Tn. Our theorem is a generalization of Skopenkov's description of graphs basically embeddable into R2, and our proofs is a (non-trivial) extension of that one.
Our methods for book embeddings were inspired by basic embeddings of graphs @cite_5 and by a reduction of the topological classification for graphs embedded in @math to word problems in finitely presented semigroups @cite_0 .