Three-page encoding and complexity theory for spatial graphs.

Abstract

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.