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We examine the topological entanglements of polygons confined to a lattice tube and under the influence of an external tensile force f. The existence of the limiting free energy for these so-called stretched polygons is proved and then, using transfer matrix arguments, a pattern theorem for stretched polygons is proved. Note that the tube constraint allows us to prove a pattern theorem for any arbitrary value of f, while without the tube constraint it has so far only been proved for large values of f. The stretched polygon pattern theorem is used first to show that the average span per edge of a randomly chosen n-edge stretched polygon approaches a positive value, non-decreasing in f, as n → ∞. We then show that the knotting probability of an n-edge stretched polygon confined to a tube goes to one exponentially as n → ∞. Thus as n → ∞ when polygons are influenced by a force f, no matter its strength or direction, topological entanglements, as defined by knotting, occur with high probability.

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.

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