Knots and links in certain spatial complete graphs

Abstract

We give a spatial representation of the complete graph Kn which contains exactly (n6) Hopf links consisting of pairs of disjoint 3-cycles, where (n6) is the minimum number of pairs of disjoint 3-cycles forming a non-split link contained in any spatial representation of Kn. Furthermore, we show that the above spatial representation of Kn contains exactly (n7) trefoil knots consisting of 7-cycles, where (n7) is the minimum number of 7-cycles forming a non-trivial knot contained in any spatial representation of Kn. © 1996 Academic Press, Inc.