Dynnikov three-page diagrams of spatial 3-valent graphs (Q5959540)

The author extends the approach of \textit{I. A. Dynnikov} for the algebraic classification of links [Funct. Anal. Appl. 34, 24-32 (2000; Zbl 0958.57006)] to the case of \(3\)-valent graphs embedded in \(\mathbb R^3\). The motivations for this study are not only topological but also related to molecular biology. An abstract semigroup \(Stg\) is defined by explicitly giving (finitely many) generators and relations. The first result of the paper states that any spatial graph can be encoded by an element of \(Stg\). The idea of the proof is to embed the graph in a three-page book, so that the \(3\)-valent graph can be recovered by writing out the types of vertices on the axis of the book. The main result is that the center of \(Stg\) is in one-to-one correspondence with the isotopy classes of \(3\)-valent graphs embedded in \(\mathbb R^3\).