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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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References

REFERENCES

1Burde, G. and Zieschang, H.. Knots (de Gruyter, 1985).Google Scholar
2Conway, J. H. and Gordon, C. McA.. Knots and links in spatial graphs. J. Graph Theory 7 (1983), 445453.CrossRefGoogle Scholar
3Ernst, C. and Sumners, D. W.. The growth of the number of prime knots. Math. Proc. Cambridge Philos. Soc. 102 (1987), 303315.CrossRefGoogle Scholar
4Flapan, E. and Weaver, N.. Intrinsic ehirality of complete graphs. Preprint (1990).Google Scholar
5de Gennes, P. G.. Tight knots. Macromolecules 17 (1984), 703704.CrossRefGoogle Scholar
6Hammersley, J. M.. Percolation processes II. The connective constant. Proc. Cambridge Philos. Soc. 53 (1957), 642645.CrossRefGoogle Scholar
7Hammersley, J. M.. The number of polygons on a lattice. Proc. Cambridge Philos. Soc. 57 (1961), 516523.CrossRefGoogle Scholar
8Hammersley, J. M. and Welsh, D. J. A.. Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. Oxford Ser. (2) 13 (1962), 108110.CrossRefGoogle Scholar
9de la Harpe, P., Kervaire, M. and Weber, C.. On the Jones polynomial. Enseign. Math. (2) 32 (1986), 271335.Google Scholar
10Jones, V. F. R.. A polynomial invariant for knots. Bull. Amer. Math. Soc. 12 (1985), 103111.CrossRefGoogle Scholar
11Kauffman, L. H.. State models and the Jones polynomial. Topology 26 (1987), 395407.CrossRefGoogle Scholar
12Kesten, H.. On the number of self-avoiding walks. J. Math. Phys. 4 (1963), 960969.CrossRefGoogle Scholar
13Kuratowski, G.. Sur le probleme des courbes gauche en topologie. Fund. Math. 15 (1930), 271283.CrossRefGoogle Scholar
14Lickorish, W. B. R.. Polynomials for links. Bull. London Math. Soc. 20 (1988), 558588.CrossRefGoogle Scholar
15Lickorish, W. B. R.. The unknotting number of a classical knot. Contemp. Math. 44 (1985), 117121.CrossRefGoogle Scholar
16Madras, N. and Sokal, A. D.. Non-ergodicity of local, length-conserving Monte Carlo algorithms for the self-avoiding walk. J. Statist. Phys. 47 (1987), 573595.CrossRefGoogle Scholar
17Madras, N., Soteros, C. E. and Whittington, S. G.. Statistics of lattice animals. J. Phys. A 21 (1988), 46174635.CrossRefGoogle Scholar
18Mason, W. K.. Homeomorphic continuous curves in 2-space are isotopic in 3-space. Trans. Amer. Math. Soc. 142 (1969), 269290.Google Scholar
19Morton, H. R.. Seifert circles and knot polynomials. Math. Proc. Cambridge Philos. Soc. 99 (1986), 107109.CrossRefGoogle Scholar
20Murasugi, K.. Jones polynomials and classical conjectures in knot theory. Topology 26 (1987), 187194.CrossRefGoogle Scholar
21Nakanishi, Y.. A note on unknotting number. Math. Sern. Notes Kobe Univ. 9 (1981), 99108.Google Scholar
22Pippenger, N.. Knots in random walks. Discrete Appl. Math. 25 (1989), 273278.CrossRefGoogle Scholar
23Schubert, H.. Die eindeutige Zerlegbarkeit eines Knoten in Primknoten. Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl. 3 (1949), 57104.Google Scholar
24Schubert, H.. ber eine numerische Knoteninvariante. Math. Z. 61 (1954), 245288.CrossRefGoogle Scholar
25Sumners, D. W. and Whittington, S. G.. Knots in self-avoiding walks. J. Phys. A 21 (1988), 16891694.CrossRefGoogle Scholar
26Thistlethwaite, M. B.. Knot tabulations and related topics. In Aspects of Topology in Memory of Hugh Dowker, 19121982, London Math. Soc. Lecture Notes no. 93 (Cambridge University Press, 1985), pp. 176.Google Scholar
27Thistlethwaite, M. B.. A spanning tree expansion of the Jones polynomial. Topology 26 (1987), 297310.CrossRefGoogle Scholar
28Wasserman, S. A. and Cozzarelli, N. R.. Biochemical topology: Application to DNA recombination and replication. Science 232 (1986), 951960.CrossRefGoogle ScholarPubMed
29Yamada, S.. The minimum number of Seifert circles equals the braid index of a link. Invent. Math. 89 (1987), 347356.CrossRefGoogle Scholar