Lattice Knots Research Articles

Vertex distortion detects the unknot

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.

Vertex distortion of lattice knots

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.

Reply to "Comment on 'Osmotic pressure of compressed 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).

Comment on “Osmotic pressure of compressed lattice knots”

In a recent paper, E. J. Janse van Rensburg has presented computational data enumerating the conformations of closed circular self-avoiding lattice polymers with knots confined in a cubic box, and claimed to have observed a negative osmotic pressure in the system. The purpose of this comment is to state that osmotic pressure by a self-avoiding polymer, knotted or otherwise, is positive, which means a polymer pushes confining walls outwards, and the statement of the opposite is a mistake.

The free energy of compressed lattice knots

A compressed knotted ring polymer in a confining cavity is modelled by a knotted lattice polygon confined in a cube in . The GAS algorithm (Janse van Rensburg and Rechnitzer 2011 J. Knot Theol. Raman 20 1145–71) is used to sample lattice polygons of fixed knot type in a confining cube and to estimate the free energy of confined lattice knots. Lattice polygons of knot types the unknot, the trefoil knot, and the figure eight knot, are sampled and the free energies are estimated as functions 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 (Flory 1942 J. Chem. Phys. 10 51–61; Huggins 1942 J. Phys. Chem. 46 151–8) is used to model the free energy as a function of monomer concentration. The Flory interaction parameter of knotted lattice polygons in is also estimated.

The free energy of compressed lattice knots

Abstract 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.

Osmotic pressure of compressed lattice knots

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.

Neutronographic characterization of II‐VI cubic crystals highly doped by 3d ions: on possible tendencies to structure rearrangements in the sphalerite crystal lattice

AbstractFine features of the crystal structure of Zn0.9Ni0.1S, Zn0.9V0.1Se, Zn0.95Fe0.05Se semiconductor cubic single crystals have been characterized by neutron scattering at room temperature. It has been revealed that neutron‐diffraction scans of the single crystals, in addition to intense Bragg reflections of face‐centered cubic phase, contain a complex system of diffuse scattering maxima, included superstructure reflections. The superstructure diffuse maxima are found placed on reciprocal lattice knots corresponding with the wave vectors q = (1/3 1/3 1/3) 2π/ac (ac is the cubic lattice parameter), which is interpreted as a manifestation of pretransition state. It is proposed that the revealed destabilized structure state means pretransition of reconstructive structure transformation from the cubic phase to the hexagonal phase. Strong perturbations caused by 3d ions doping lead to a collective response of the cubic lattice of the II‐VI matrices, displacing atomic planes in shears, and tend to induce the transformation. Such a complicated structure state determines nonuniform deformation field supporting possibilities to form long‐wave superstructures. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The entropic pressure of lattice knots

.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.

Upper bounds on the minimum length of cubic lattice knots

Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot. In this paper, we consider every knot in the cubic lattice. In particular, the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice. Diao introduced this term (he used ‘minimum edge number’ instead) and proved that the minimum length of the trefoil knot 31 is 24. Also the minimum lengths of the knots 41 and 51 are known to be 30 and 34, respectively. In this paper we find a general upper bound of the minimum length of a nontrivial knot K, except the trefoil knot, in terms of the minimum crossing number c(K). The upper bound is . Moreover, if K is a non-alternating prime knot, then the upper bound is . Our work can be considered a direct consequence of the results obtained by the authors in Hong et al (2012 arXiv:1209.0048). Furthermore, if K is (n + 1, n)-torus knot, then the upper bound is .

Lattice knots in a slab

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.

The compressibility of minimal lattice knots

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.

Size-consistent variational approaches to nonlocal pseudopotentials: Standard and lattice regularized diffusion Monte Carlo methods revisited

We propose improved versions of the standard diffusion Monte Carlo (DMC) and the lattice regularized diffusion Monte Carlo (LRDMC) algorithms. For the DMC method, we refine a scheme recently devised to treat nonlocal pseudopotential in a variational way. We show that such scheme-when applied to large enough systems-maintains its effectiveness only at correspondingly small enough time-steps, and we present two simple upgrades of the method which guarantee the variational property in a size-consistent manner. For the LRDMC method, which is size-consistent and variational by construction, we enhance the computational efficiency by introducing: (i) an improved definition of the effective lattice Hamiltonian which remains size-consistent and entails a small lattice-space error with a known leading term and (ii) a new randomization method for the positions of the lattice knots which requires a single lattice-space.

Bounds for the minimum step number of knots in the simple cubic lattice

Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numerically, the minimum length (also called minimum step number) needed to form a particular knot in the simple cubic lattice. Our analytical work is based on improvement of a method introduced by Diao to enumerate conformations of a given knot type for a fixed length. This method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 (which is 24) to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34, respectively. Using an independent method based on the BFACF algorithm, we provide a complete list of numerical estimates (upper bounds) of the minimum step numbers for prime knots up to ten crossings, which are improvements over current published numerical results. We enumerate all minimum lattice knots of a given type and partition them into classes defined by BFACF type 0 moves.

Thoughts on lattice knot statistics

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.

ON THE RELATIONSHIP BETWEEN MINIMAL LATTICE KNOTS AND MINIMAL CUBE KNOTS

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.

Thermodynamics and topology of disordered systems: Statistics of the random knot diagrams on finite lattices

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.

The Curvature of Lattice Knots

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.

The complexity of lattice knots

A family of polygonal knots K n on the cubical lattice is constructed with the property that the quotient of length L( K n ) over the crossing number Cr( K n ) approaches zero as L approaches infinity. More precisely Cr(K n) = O(L(K n) 4 3 ) . It is shown that this construction is optimal in the sense that for any knot K on the cubical lattice with length L and Cr crossings Cr ⩽ 3.2L 4 3 .

The Writhe of Knots in the Cubic Lattice

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.