Entanglement complexity of graphs in <i>Z</i>³

Abstract

Abstract In this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z 3 . What is the probability that such a randomly chosen embedding is knotted? What is the probability that the embedding contains a particular knot? What is the expected complexity of the knot? To what extent can these questions also be answered for a graph of a given homeomorphism type? We use a pattern theorem due to Kesten 12 to prove that almost all embeddings in Z 3 of a sufficiently long closed curve contain any given knot. We introduce the idea of a good measure of knot complexity. This is a function F which maps the set of equivalence classes of embeddings into 0, ). The F measure of the unknot is zero, and, generally speaking, the more complex the prime knot decomposition of a given knot type, the greater its F measure. We prove that the average value of F diverges to infinity as the length ( n ) of the embedding goes to infinity, at least linearly in n . One example of a good measure of knot complexity is crossing number. Finally we consider similar questions for embeddings of graphs. We show that for a fixed homeomorphism type, as the number of edges n goes to infinity, almost all embeddings are knotted if the homeomorphism type does not contain a cut edge. We prove a weaker result in the case that the homeomorphism type contains at least one cut edge and at least one cycle.