Knot theory of spatial graphs

Abstract

The topological study of spatial graphs is considered to be a natural extension of knot theory, although it has not been paid much attention until quite recently. In this chapter, we regard two notions on “equivalence” of graphs. The first one is a notion naturally extending positive-equivalence of links and is called equivalence. The second one is a notion which is useful when we study the exterior of a spatial graph and is called neighborhood-equivalence. Since the importance of the first concept is motivated by recent developments in molecular chemistry, we devote the first section to some comments on the topology of molecules. In 15.2 we discuss some results on the first notion, and in 15.3 some results on the second notion, including an explanation of recent developments on the tunnel number.