The 3-move and Knotted 4-valent Graphs in 3-space

A topological graph is a one-dimensional complex consisting of finitely many 0-cells (vertices) and finitely many 1-cells (edges and loops). In [7], Kauffman proved that piecewise linear ambient isotopy of a piecewise linear embedding of a topological graph in Euclidean 3-space R3 or 3-sphere 3, referred simply a knotted graph, is generated by a set of diagrammatic local moves (see Fig. 1) that generalize the Reidemeister moves for diagrams of classical links. This gives a complete combinatorial description of the topology of graphs in three dimensional space. Throughout this paper, all spaces and maps are in piecewise linear category and we speak of 3-space in referring to either R3 or 3 = R3 ∪ {∞}. A method for producing invariants of knotted graphs in 3-space is to associate a collection of links to the knotted graph [7, 13] and also a polynomial invariant for knotted graphs is developed [16]. On the other hand, ambient isotopy of knotted graphs is rather complicated by the fact that the generalized Reidemeister move (V) (see Fig. 1) creates or destroys arbitrary braiding at a vertex and so it is not easy to define non trivial invariants of the braiding move (V). For this reason, many authors turned their attention to restrict the valency of vertices and the allowed movement in the neighborhoods of vertices. This makes the construction of invariants of such graphs rather easier [1, 5, 7, 8, 13, 14, 15, 18]. The purpose of this paper is to introduce a method for obtaining invariants of the braiding move (V) and consequently producing invariants of knotted 4-valent graphs, by using the 3-move for knots and links. This paper is organized as follows. Section 2 contains fundamental concepts for graph embeddings in 3-space. In Section 3 we associate a collection of knots and links to a knotted 4-valent graph in 3-space and show that the 3-equivalent class of the collection is an invariant of the knotted 4-valent graphs. In Section 4 we construct new 3-move invariants by using Kauffman bracket polynomial and show that this 3-move invariant gives a useful way to distinguish knotted 4-valent graphs in 3-space.