Topology and Geometry in Polymer Science pp 101-133 | Cite as

# Knots in Graphs in Subsets of *Z*^{3}

## Abstract

The probability that an embedding of a graph in *Z* ^{3} is knotted is investigated. For any given graph (embeddable in *Z* ^{3}) without cut edges, it is shown that this probability approaches 1 at an exponential rate as the number of edges in the embedding goes to infinity. Furthermore, at least for a subset of these graphs, the rate at which the probability approaches 1 does not depend on the particular graph being embedded. Results analogous to these are proved to be true for embeddings of graphs in a subset of *Z* ^{3} bounded by two parallel planes (a slab).

In order to investigate the knotting probability of embeddings of graphs in a rectangular prism (an infinitely long rectangular tube in *Z* ^{3}), a pattern theorem for self-avoiding polygons in a prism is proved. From this it is possible to prove that for any given eulerian graph, the probability that an embedding of the graph in a prism is knotted goes to 1 as the number of edges in the embedding goes to infinity. Then, just as for *Z* ^{3}, there is at least a subset of these graphs for which the rate that this probability approaches 1 does not depend on the particular graph.

Similar results are shown to hold in cases where restrictions are placed on the number of edges per branch in a graph embedding.

## Keywords

knots graph embeddings branched polymer simple cubic lattice## Preview

Unable to display preview. Download preview PDF.

## References

- [1]D.W. Sumners and S.G. Whittington, Knots in self-avoiding walks, J. Physics A: Math. Gen., 21 (1988), pp. 1689–1694.MathSciNetCrossRefMATHGoogle Scholar
- [2]N. Pippenger,
*Knots in random walks*, Discrete Applied Math., 25 (1989), pp. 273–278.MathSciNetCrossRefMATHGoogle Scholar - [3]C.E. Soteros, D. W. Sumners and S. G. Whittington
*Entanglement complexity of graphs**in**Z*^{3}, Proc. Camb. Phil. Soc., 111 (1992), pp. 75–91.MathSciNetCrossRefMATHGoogle Scholar - [4]M.C. Tesi, E.J. Janse Van Rensburg, E. Orlandini and S.G. Whittington,
*Knot probability for lattice polygons in confined geometries*, J. Phys A: Math. Gen., 27 (1994), pp. 347–360.CrossRefMATHGoogle Scholar - [5]C.E. Soteros,
*Random knots in uniform branched polymers*, Mathematical Modelling and Scientific Computing, 2, Section B (1993), pp. 747–752. (Proceedings of the Eighth International Conference on Mathematical and Computer Modelling. April 1–4,1991, University of Maryland, College Park, Maryland)Google Scholar - [6]J.M. Hammersley and K.W. Morton
*Poor man’s Monte**Carlo*, J. R. Stat. Soc. B. 16 (1959), pp. 23–38.MathSciNetGoogle Scholar - [7]
- [8]J.M. Hammersley,
*The*number*of polygons on a lattice*, Proc. Camb. Phil. Soc., 57 (1961), pp. 516–523.MathSciNetCrossRefMATHGoogle Scholar - [9]J.M. Hammersley and S.G. Whittington,
*Self-avoiding walks in wedges*, J. Phys. A: Math. Gen., 18 (1985), pp. 101–111.MathSciNetCrossRefGoogle Scholar - [10]C.E. Soteros,
*Adsorption of uniform lattice animals with specified topology*, J. Phys. A: Math. Gen., 25 (1992), pp. 3153–3173.MathSciNetCrossRefMATHGoogle Scholar - [11]C.E. Soteros, D.W. Sumners and S.G. Whittington,
*Linked p-spheres in Zd*, (1997),*in preparation*Google Scholar - [12]J.H. Conway and C.Mca. Gordon,
*Knots and links in spatial graphs*, J. Graph Theory, 7 (1983), pp. 445–453.MathSciNetCrossRefMATHGoogle Scholar - [13]H. Kesten,
*On the number of self-avoiding walks*, J. Math. Phys., 4 (1963), pp. 960–969.MathSciNetCrossRefMATHGoogle Scholar - [14]M.C. Tesi, E.J. Janse Van Rensburg, E. Orlandini and S.G. Whittington,
*Topological entanglement complexity of polymer chains in confined goemetries*,(1997), IMA Vol. in Math, and its Appl. (Springer-Verlag) this volume.Google Scholar - [15]S.G. Whittington and C.E. Soteros,
*Uniform branched polymers in confined geometries*, Macromolecular Reports, A29(Suppl. 2) (1992), pp. 195–199.Google Scholar - [16]C.E. Soteros and S.G. Whittington,
*Lattice models of branched polymers: effects of geometrical constraints*, J. Phys A: Math. Gen., 22 (1989), pp. 5259–5270.MathSciNetCrossRefMATHGoogle Scholar - [17]S.G. Whittington,
*Statistical Mechanics of Polymer Solutions and Polymer Adsorption*, Advances in Chemical Physics, 51 (1982), pp. 1–48.CrossRefGoogle Scholar - [18]S.E. Alm and S. Janson,
*Random self-avoiding walks on one-dimensional lattices*, Commun. Statist.-Stochastic Models, 6 (1990), pp. 169 212.MathSciNetGoogle Scholar - [19]H.H. Schaefer, Banach
*Lattices and Positive Operators*, (1974), (Berlin; New York: Springer-Verlag).MATHGoogle Scholar - [20]C.E. Soteros, (1997),
*in preparation*.Google Scholar