For any knotted planar graph H in R 3 , there exist a planar graph G and its regular projection G /R 2 such that every diagram obtained from G contains a subdiagram that represents H.
1998 Academic Press
1. Introduction
Let G be a finite graph. We denote the vertex set and the edge set of G by V(G) and E(G), respectively. A cycle is a loop or a simple connected graph which is homeomorphic to a circle. A path is a simple connected graph which is homeomorphic to a closed line segment. A spatial embedding of G is an embedding g: G ร R 3 of G into the three-dimensional Euclidian space R 3 , and its image G g = g(G) is called a spatial graph. If it consists of a single cycle or a disjoint union of cycles, then it is called a knot or a link. A graph is called planar if it has an embedding into R 2 . A regular projection of a graph is its drawing on the plane whose multiple points are only finitely many double points of edges. If we give over'' and under'' information at each double point to a regular projection of a graph, then we obtain a spatial graph (near the plane) and we call it a diagram of a spatial graph (Fig. 1.1). We note that every diagram in this paper is obtained in this way. A diagram is useful for studying a spatial graph.
Recently, there are many works on Ramsey-type theorems for spatial graphs [4 7, 11]. These follow the celebrated paper of Conway and Gordon [2]. They showed the following theorem.
Theorem 1.1 (Conway and Gordon [2]). Every spatial embedding of K 7 contains a nontrivial knot.