Invariants of Graphs in Three-Space

By associating a collection of knots and links to a graph in threedimensional space, we obtain computable invariants of the embedding type of the graph. Two types of isotopy are considered: topological and rigid-vertex isotopy. Rigid-vertex graphs are a category mixing topological flexibility with mechanical rigidity. Both categories constitute steps toward models for chemical and biological networks. We discuss chirality in both rigid and topological contexts. This paper introduces methods for producing topological invariants of graphs embedded in three-dimensional space. For this study, two types of graphs are considered: graphs with rigid vertices and graphs with nonrigid vertices. Our methods are strongest for graphs with rigid vertices. The primary method in either case is to associate a collection of knots and links to thegraph such that, for the appropriate notion of isotopy, this collection (the isotopy class of the collection) is an invariant of the isotopy type of the graph. One may proceed purely geometrically, or apply knot theoretic invariants to the collection. This method is simple and powerful. By not opting too early for the extra structure of a polynomial invariant, we can often directly use the topology to create a minimal solution. Nevertheless, it turns out to be particularly fruitful to use the new polynomial invariants of Jones and others (see [3, 5, 6, 7, 9, 10, 131). Also, by combining knot theoretic invariants, we [lo] have produced nontrivial graph-polynomial invariants in the case of 4-valent rigid vertex graphs. The latter invariants generalize the known two-variable polynomial invariants of knots and links (Homfly and Kauffman polynomials). These generalizations are joint work of the author and Pierre Vogel [lo]. I shall show here ($4) how these polynomials are related to the point of view involving a collection of links associated with the graph. The paper is organized as follows: In $2 we define ambient isotopy for nonrigid (topological) vertices and prove that piecewise linear ambient isotopy is generated diagrammatically by a set of moves (Figure 1) that generalize the Reidemeister moves). $2 then shows Received by the editors December 1, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. @ 1989 Amencan Mathemat~cal Soclely 0002.9947189 $ l 00 + $ 2 5 per page