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Entanglement complexity of graphs in Z3

Published online by Cambridge University Press:  24 October 2008

C. E. Soteros ,
D. W. Sumners  and
S. G. Whittington
C. E. Soteros
Affiliation:
Department of Mathematics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 0W0, Canada
D. W. Sumners
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306-3027, U.S.A.
S. G. Whittington
Affiliation:
Department of Chemistry, University of Toronto, Toronto, Ontario M5S 141, Canada
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Abstract

In this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z3. 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 Z3 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 1 , January 1992 , pp. 75 - 91
Copyright
Copyright © Cambridge Philosophical Society 1992

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