A Note on Lattice Knots

The free energy of a model of uniformly weighted lattice knots of length n and knot type K confined to a lattice cube of side length L-1 is given by F_{L}(ϕ)=-1/Vlogp_{n,L}(K), where V=L^{3} and where ϕ=n/V is the concentration of monomers of the lattice knot in the confining cube. The limiting free energy of the model is F_{∞}(ϕ)=lim_{L→∞}F_{L}(ϕ) and the limiting osmotic pressure of monomers leaving the lattice knot to become solvent molecules is defined by Π_{∞}(ϕ)=ϕ^{2}d/dϕ[F_{∞}(ϕ)/ϕ]. I show that, under very mild assumptions, the functions P_{L}(ϕ)=ϕ^{2}d/dϕ[F_{L}(ϕ)/ϕ]|_{n} and Π_{L}(ϕ)=ϕ^{2}d/dϕ[F_{L}(ϕ)/ϕ]|_{L} are finite-size approximations of Π_{∞}(ϕ).

A numerical simulation shows that the osmotic pressure of compressed lattice knots is a function of knot type, and so of entanglements. The osmotic pressure for the unknot goes through a negative minimum at low concentrations, but in the case of nontrivial knot types 3_{1} and 4_{1} it is negative for low concentrations. At high concentrations the osmotic pressure is divergent, as predicted by Flory-Huggins theory. The numerical results show that each knot type has an equilibrium length where the osmotic pressure for monomers to migrate into and out of the lattice knot is zero. Moreover, the lattice unknot is found to have two equilibria, one unstable, and one stable, whereas the lattice knots of type 3_{1} and 4_{1} have one stable equilibrium each.

The (isothermic) compressibility of lattice knots can be examined as a model of the effectsof topology and geometry on the compressibility of ring polymers. In this paper, thecompressibilities of minimal length lattice knots in the simple cubic, face centeredcubic and body centered cubic lattices are determined. Our results show thatthe compressibility is generally not monotonic, but in some cases increases withpressure. Differences in the compressibility for different knot types show that thetopology is a factor determining the compressibility of a lattice knot, and differencesbetween the three lattices show that the compressibility is also a function of thegeometry.

Lattice knot statistics, or the study of knotted polygons in the cubic lattice, gained momentum in 1988 when the Frisch-Wasserman-Delbruck conjecture was proven by Sumners and Whittington (J Phys A Math Gen 21:L857–861, 1988), and independently in 1989 by Pippenger (Disc Appl Math 25:273–278, 1989). In this paper, aspects of lattice knot statistics are reviewed. The basic ideas underlying the study of knotted lattice polygons are presented, and the many open problem are posed explicitly. In addition, the properties of knotted polygons in a confining slab geometry are explained, as well as the Monte Carlo simulation of knotted polygons in \({{\mathbb Z}^3}\) and in a slab geometry. Finally, the mean behaviour of lattice knots in a slab are discussed as a function of the knot type.

The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the [Formula: see text]-norm. Inspired by Gromov, Pardon and Blair–Campisi–Taylor–Tomova, we show that results about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and there are minimal lattice-stick number knot conformations with arbitrarily high distortion.

.The entropic pressure in the vicinity of a cubic lattice knot is examined as a model of the entropic pressure near a knotted ring polymer in a good solvent. A model for the scaling of the pressure is developed and this is tested numerically by sampling lattice knots using a Monte Carlo algorithm. Good agreement is found between scaling predictions and numerical experiments.

IOP Publishing has withdrawn this article upon the author's request due to several issues with the mathematical content. The osmotic pressure of monomers in a knotted ring polymer in a confining cavity is modelled by a lattice polygon confined in a cube in ${\mathbb Z}^3$. These polygons can be knotted and are called lattice knots. In this paper the GAS algorithm \cite{JvRR11} is used to estimate the free energy of lattice knots of knot types the unknot, the trefoil knot, and the figure eight knot, as a function of the concentration of monomers in the confining cube. The data show that the free energy is a function of knot type at low concentrations, and (mean-field) Flory-Huggins theory \cite{Flory42,H42} is used to model the free energy as a function of monomer concentration. The Flory interaction parameter of lattice polygons in ${\mathbb Z}^3$ is also estimated. At critical values of the concentration the osmotic pressure may vanish, and these critical concentrations, suitably rescaled, is dependent on knot type.

The writhe of a knot in the simple cubic lattice [Formula: see text] can be computed as the average linking number of the knot with its pushoffs into four non-antipodal octants. We use a Monte Carlo algorithm to generate a sample of lattice knots of a specified knot type, and estimate the distribution of the writhe as a function of the length of the lattice knots. If the expected value of the writhe is not zero, then the knot is chiral. We prove that the writhe is additive under concatenation of lattice knots and observe that the mean writhe appears to be additive under the connected sum operation. In addition we observe that the mean writhe is a linear function of the crossing number in certain knot families.

In this paper the number and lengths of minimal length lattice knots confined to slabs of width L are determined. Our data on minimal length verify the recent results by Ishihara et al for the similar problem, except in a single case, where an improvement is found. From our data we construct two models of grafted knotted ring polymers squeezed between hard walls, or by an external force. In each model, we determine the entropic forces arising when the lattice polygon is squeezed by externally applied forces. The profile of forces and compressibility of several knot types are presented and compared, and in addition, the total work done on the lattice knots when they are squeezed to a minimal state is determined.

A result of Milnor [1] states that the infimum of the total curvature of a tame knot K is given by 2πμ(K), where μ(K) is the crookedness of the knot K. It is also known that μ(K)=b(K), where b(K) is the bridge index of K [2]. The situation appears to be quite different for knots realised as polygons in the cubic lattice. We study the total curvature of lattice knots by developing algebraic techniques to estimate minimal curvature in the cubic lattice. We perform simulations to estimte the minimal curvature of lattice knots, and conclude that the situation is very different than for tame knots in ℛ3.

In Chapter 1 the self-avoiding walk connective constant, growth constant and subadditivity are introduced. Concatenation and the supermultiplicativity of lattice polygons are discussed. Scaling, critical exponents and the Flory theory of self-avoiding walks, as well as scaling and hyperscaling relations for self-avoiding walks, are reviewed. Walk and polygon generating functions are presented and tadpoles, thetas, figure eights and dumbbells are discussed. Lattice knots and Flory theory for lattice knots are covered in the final part of the chapter.

The statistical properties of random lattice knots, the topology of which is determined by the algebraic topological Jones-Kauffman invariants, was studied by analytical and numerical methods. The Kauffman polynomial invariant of a random knot diagram was represented by a partition function of the Potts model with a random configuration of ferro-and antiferromagnetic bonds, which allowed the probability distribution of the random dense knots on a flat square lattice over topological classes to be studied. A topological class is characterized by the highest power of the Kauffman polynomial invariant and interpreted as the free energy of a q-component Potts spin system for q→∞. It is shown that the highest power of the Kauffman invariant correlates with the minimum energy of the corresponding Potts spin system. The probability of the lattice knot distribution over topological classes was studied by the method of transfer matrices, depending on the type of local junctions and the size of the flat knot diagram. The results obtained are compared to the probability distribution of the minimum energy of a Potts system with random ferro-and antiferromagnetic bonds.

Lattice knots have been studied in recent years, especially in ℤ3 and with respect to how many edges are required to form a knot. Knots formed from cubes have also been investigated for their ability to tessellate space. In this article, we demonstrate that there is a relationship between the minimum number of edges required to form a lattice knot and the minimum number of cubes required to form the same kind of knot. We further investigate the relationship in the face-centered cubic lattice.

The first two authors introduced vertex distortion of lattice knots and showed that the vertex distortion of the unknot is 1. It was conjectured that the vertex distortion of a knot class is 1 if and only if it is trivial. We use Denne and Sullivan’s lower bound on Gromov distortion to bound the vertex distortion of non-trivial lattice knots. This bounding allows us to conclude that a knot class has vertex distortion 1 if and only if it is trivial. We also show that vertex distortion does not have a universal upper bound and provide a vertex distortion calculator.

In this paper we examine the relative knotting probabilities in a lattice model of ring polymers confined in a cavity. The model is of a lattice knot of size n in the cubic lattice, confined to a cube of side length L and with volume V=(L+1)^{3} sites. We use Monte Carlo algorithms to estimate approximately the number of conformations of lattice knots in the confining cube. If p_{n,L}(K) is the number of conformations of a lattice polygon of length n and knot type K in a cube of volume L^{3}, then the relative knotting probability of a lattice polygon to have knot type K, relative to the probability that the polygon is the unknot (the trivial knot, denoted by 0_{1}), is ρ_{n,L}(K/0_{1})=p_{n,L}(K)/p_{n,L}(0_{1}). We determine ρ_{n,L}(K/0_{1}) for various knot types K up to six crossing knots. Our data show that these relative knotting probabilities are small over a wide range of the concentration φ=n/V of monomers for values of L≤12 so that the model is dominated by unknotted lattice polygons. Moreover, the relative knot probability increases with φ along a curve that flattens as the Hamiltonian state is approached.