We consider self-avoiding polygons in a restricted geometry, namely an infinite L × M tube in Z 3 . These polygons are subjected to a force f , parallel to the infinite axis of the tube. When f > 0 the force stretches the polygons, while when f < 0 the force is compressive. We obtain and prove the asymptotic form of the free energy in both limits f → ±∞. We conjecture that the f → −∞ asymptote is the same as the limiting free energy of "Hamiltonian" polygons, polygons which visit every vertex in a L × M × N box. We investigate such polygons, and in particular use a transfer-matrix methodology to establish that the conjecture is true for some small tube sizes.Dedicated to Anthony J. Guttmann on the occasion of his 70 th birthday. arXiv:1604.07465v1 [math-ph]

Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either "local" or "non-local", based on a standard approach for measuring the "size" of a knot within a knotted polymer chain. The method involves associating knot-types to subarcs of the chain, and then identifying a knotted subarc with minimal arclength; this arclength is then the knot-size. If the resulting knot-size is small relative to the whole length of the chain, then the knot is considered to be localised or "local"; otherwise, it is "non-local". Using this definition, we establish that all but exponentially few sufficiently long self-avoiding polygons (closed chains) in a tubular sublattice of the simple cubic lattice are "non-locally" knotted. This is shown to also hold for the case when the same polygons are subject to an external tensile force, as well as in the extreme case when they are as compact as possible (no empty lattice sites). We also provide numerical evidence for small tube sizes that at equilibrium non-local knotting is more likely than local knotting, regardless of the strength of the stretching or compressing force. The relevance of these results to other models and recent experiments involving DNA knots is also discussed.

We study several related models of self-avoiding polygons in a tubular subgraph of the simple cubic lattice, with a particular interest in the asymptotics of the knotting statistics. Polygons in a tube can be characterised by a finite transfer matrix, and this allows for the derivation of pattern theorems, calculation of growth rates and exact enumeration. We also develop a static Monte Carlo method which allows us to sample polygons of a given size directly from a chosen Boltzmann distribution.Using these methods we accurately estimate the growth rates of unknotted polygons in the 2 × 1 × ∞ and 3 × 1 × ∞ tubes, and confirm that these are the same for any fixed knot-type K. We also confirm that the entropic exponent for unknots is the same as that of all polygons, and that the exponent for fixed knot-type K depends only on the number of prime factors in the knot decomposition of K. For the simplest knot-types, this leads to a good approximation for the polygon size at which the probability of the given knot-type is maximized, and in some cases we are able to sample sufficiently long polygons to observe this numerically.Dedicated to Stuart Whittington on the occasion of his 75 th birthday. *

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