Knotted Symmetric Graphs

For a knotted graph in S3 we define the vertex constant group, a quotient of the fundamental group of the complement. For planar graphs the group is cyclic. For graphs with periodic symmetry the group is related to the fundamental group of the branched cover of S3 branched over knots contained in the quotient of the graph under the symmetry. These tools are used to demonstrate that a large family of knotted graphs are not planar. This note presents a new tool for distinguishing knotted graphs in S3. To each such graph F we associate a group, called the vertex constant group; for connected planar graphs the vertex constant group is finite cyclic. Our first application is to offer a new proof that all elements of a particular family of knotted graphs, denoted on, are nontrivial. This family of graphs was introduced by Suzuki [Su], expanding on an example of Kinoshita [Ki]; it is of particular interest as a test case for new methods since each element is Brunnian-that is, every subgraph is planar. Suzuki proved that the on are nontrivial using an algebraic argument in which he showed that the fundamental group of the complement is not free; Scharlemann [S] has described a more geometric approach, reducing the problem to a calculation in the braid group. Our second application is to demonstrate that the on represents a special case of a large class of knotted graphs, all of which will be proved nontrivial without any explicit calculation. Neither of the approaches of [S, Su] nor apparently any other known techniques apply to the entire class. (See [Kl, K2, T, Y] for recent progress in the study of knotted graphs.) Figure 1 illustrates the graph 65. One vertex has been placed at the point at infinity to highlight the periodic symmetry of the graph. Similarly, on has two vertices joined by n edges. Note that there is a period n transformation, T, of S3 leaving n, invariant. This action determines a branched cover of S3, the image of in, is an arc, K. Figure 2 illustrates K along with the branch set, described as the union of two arcs, B1 and B2. The union of B2 and K is a trefoil knot. We will see that if the edges of on are replaced with any other periodic family of edges, the graph continues to be nontrivial, as long as the union of B2 and K forms a nontrivial knot in S3. Received by the editors June 14, 1993. 1991 Mathematics Subject Classification. Primary 57M25; Secondary 57M60.