The 3-move and knotted 4-valent graphs in 3-space (Q1431084)

An embedding of a finite graph into Euclidean 3-space is called a \textit{knotted graph}. One of the natural ideas for the construction of invariants of knotted graphs is to consider a collection of links in knotted graphs. \textit{L. H. Kauffman} proposed a specific way to associate a collection of links to each knotted 4-valent graph [Trans. Am. Math. Soc. 311, No. 2, 697--710 (1989; Zbl 0672.57008)]. Such a collection is an invariant under rigid-vertex isotopy but not under ambient isotopy. In this paper, to resolve the defect, the authors propose an equivalence relation on links, called 3-equivalence. The main theorem of this paper is that the 3-equivalence classes of the above collection form an ambient isotopy invariant of knotted 4-valent graphs. Based on this theorem, the authors construct numeric invariants of knotted 4-valent graphs.