Christine E. Soteros
University of Saskatchewan
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Mathematical Proceedings of the Cambridge Philosophical Society | 1992
Christine E. Soteros; De Witt Sumners; S G Whittington
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.
Journal of Physics A | 1989
Christine E. Soteros; S G Whittington
The authors consider uniform brushes on subsets of the square and simple cubic lattices. They show that, in a slab geometry in three dimensions, the connective constant of a brush is identical to that of a walk in that slab. In two dimensions, in a slit geometry, the connective constant of a brush is strictly less than that of a walk. They also consider the intermediate case of a rectangular prism and show that the connective constant of a brush is strictly less than that of a walk; similar results are presented for combs, stars and polygons.
Journal of Knot Theory and Its Ramifications | 2010
M. Atapour; Christine E. Soteros; C. Ernst; S G Whittington
We consider two self-avoiding polygons (2SAPs) each of which spans a tubular sublattice of ℤ3. A pattern theorem is proved for 2SAPs, that is any proper pattern (a local configuration in the middle of a 2SAP) occurs in all but exponentially few sufficiently large 2SAPs. This pattern theorem is then used to prove that all but exponentially few sufficiently large 2SAPs are topologically linked. Moreover, we also use it to prove that the linking number Lk of an n edge 2SAP Gn satisfies limn→∞ℙ(|Lk(Gn)| ≥ f(n))=1 for any function
Journal of Physics A | 2011
Michael Szafron; Christine E. Soteros
f(n) = o(\sqrt{n})
Journal of Mathematical Chemistry | 1991
S G Whittington; Christine E. Soteros; Neal Madras
. Hence the probability of a non zero linking number for a 2SAP approaches one as the size of the 2SAP goes to infinity. It is also established that, due to the tube constraint, the linking number of an n edge 2SAP grows at most linearly in n.
Journal of Physics A | 2015
M Atapour; Christine E. Soteros; De Witt Sumners; S G Whittington
A model of local strand passage in self-avoiding polygons (SAPs) on the simple cubic lattice is investigated numerically. The polygons studied, called Θ-SAPs, contain a specific strand-passage structure, called Θ, at a fixed strand-passage site. After-strand-passage knot probabilities are estimated from a Monte Carlo study of unknotted and trefoil Θ-SAPs and the estimates are used here to investigate how knot reduction depends on the local juxtaposition structure at the strand-passage site. In particular, we observe a correlation between knot reduction and the angle of the crossing at the strand-passage site; this same angle has been shown experimentally by Neuman et al (2009 Proc. Natl Acad. Sci. USA 106 6986–91) to be important in explaining topoisomerase action on DNA. The angle of the crossing is crossing sign dependent, and, from our observations, so is knot reduction; this can be used to understand experimentally observed knot-type chirality biases.
Journal of Knot Theory and Its Ramifications | 2012
Christine E. Soteros; De Witt Sumners; S G Whittington
We consider a lattice model of branched polymers in dilute solution in which the polymer is modelled as an animal, weakly embeddable in the (simple cubic) lattice. In order to model the effect on the thermodynamic properties of changing the temperature or the quality of the solvent, we include an energy associated with the number of nearneighbour contacts between pairs of vertices of the animals. We show that the configurational free energy of the animal is a continuous function of the temperature and derive rigorous upper and lower bounds on the temperature dependence of the free energy. Finally, we comment on similarities between these results and corresponding ones for a model in which the energy is associated with the cyclomatic index of the animal.
Journal of Mathematical Chemistry | 1993
Christine E. Soteros
We consider closed 2-manifolds (2-manifolds without boundary) embedded in tubes in the hypercubic lattice, Z d, . For orientable 2-manifolds with fixed genus, , we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus. We prove a similar theorem for the non-orientable case for . If the genus is not fixed then we prove a pattern theorem and use this to show that 2-manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold. In four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non-orientable manifolds. In four dimensions, all except exponentially few 2-manifolds, both orientable and non-orientable, contain a local knotted -ball pair.
Archive | 1990
Stuart G. Whittington; Christine E. Soteros
We prove a pattern theorem for 2-spheres in tubes in Z4 and use this to prove that all except exponentially few 2-spheres in a tube in Z4 are knotted. We sketch how the argument can be applied to prove the same result for p-spheres in a tube in Zp+2, p > 2.
Israel Journal of Chemistry | 1991
S G Whittington; Christine E. Soteros
As a step towards understanding the thermodynamics of multi-branched polymer systems, we look at a lattice model of a uniform branched polymer with fixed topology interacting with a surface and ask for the free energy of the polymer as the number of monomers which compose the polymer goes to infinity. The conformations of a uniform branched polymer with fixed topology are modelled by embeddings of a graph in the simple cubic lattice. Rigorous results about this model are reviewed. The results suggest that large branched polymers in three dimensions interacting with a plane have the same free energy as large linear polymers interacting with a plane; the same is not true, however, for the corresponding two-dimensional problem where the polymer interacts with a line.