Intrinsically knotted graphs

Abstract

Abstract In 1983, Conway and Gordon [J Graph Theory 7 ( 1983 ), 445–453] showed that every (tame) spatial embedding of K 7 , the complete graph on 7 vertices, contains a knotted cycle. In this paper, we adapt the methods of Conway and Gordon to show that K 3,3,1,1 contains a knotted cycle in every spatial embedding. In the process, we establish that if a graph satisfies a certain linking condition for every spatial embedding, then the graph must have a knotted cycle in every spatial embedding. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 178–187, 2002; DOI 10.1002/jgt.10017