Physical and Numerical Models in Knot Theory

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Problem Statement and Results

• A Lemma about Polygonal Knots

• Solid Knots Made of Congruent Components

• Solid Knots of Uniform Thickness d

• References

In this paper I give a tutorial of knots energies from a mathematical point of view. A wide variety of knot energies that appeared in the literature is reconsidered, and put into context by discussing their properties. These energies include lattice knot, polygonal knot, and energies of C¹ and C² knots. In this paper I discuss the properties of a variety of knot energies proposed in the literature.

We argue that a systems of tightly knotted, linked, or braided flux tubes will have a universal mass-energy spectrum, since the length of fixed radius flux tubes depend only on the topology of the configuration. We motivate the discussion with plasma physics examples, then concentrate on the model of glueballs as knotted QCD flux tubes. Other applications will also be discussed.

Knotted segments are a natural occurence in linear chains subjected to fluctuations. We study experimentally the size and dynamics of knots in a shaken metal chain. For a given set of conditions, we find that the time it takes a knot to slip off the end of the chain depends sensitively on its topology.

We combine the global radius of curvature characterisation of knot thickness, the biarc discretisation of space curves, and simulated annealing code to compute approximations to the ideal shapes of the trefoil and figure-eight knots. The computations contain no discretisation error, and give rigorous lower bounds on thickness of the true ideal shapes. The introduction of a precise definition of a contact set for an approximately ideal shape allows us to resolve previously unobserved features. For example, in our approximations of both the ideal trefoil and figure-eight knots, local curvature is within a rather small tolerance of being active, i.e. achieving thickness, at several points along the knot.

"These vessels, whose origin is already known, wind around each other, like the twigs which form the handel of a basket. Sometime the arteries creep round the vein, like ivy round a tree; and sometimes the vein does the same round the arteries. The vein often folds itself into a kind of loops of different lengths, or forms itself into a species of knots subject to become varicious… The cord from its great length, sometimes get tied into one or more knots. These do not prevent the ordinary development of the foetus nor occasion its death as imagined by some" Baudelocque (1789).¹

We consider a thick polymer model with the purpose of providing a coarse-grained description for double-stranded DNA (dsDNA). The model is used to gain insight into how the intrinsic thickness of DNA affects the behaviour of the biomolecule subject to compaction. Our reference system is provided by the process of DNA packaging inside a viral capsid that has been recently characterized through delicate single-molecule experiments. The simplest of the two models considered here is able to provide a satisfactory quantitative accord with experimental measurement by using, as only inputs, the known dsDNA diameter and the base-pair spacing. In a second model we further incorporate a standard bending rigidity term obtaining a better agreement with experiments.

Gel electrophoresis allows to separate knotted DNA (nicked circular) of equal length according to the knot type. We present a computer simulation of knotted charged chains moving in an external electric field through grids of obstacles. Using a simple Monte-Carlo algorithm, the dependence of the electrophoretic migration of the DNA molecules on the type of knot was investigated at a low electric field regime. The results are in qualitative agreement with electrophoretic experiments done under conditions of low electric fields: in particular the electrophoretic mobility increases quasi linearly with the mean average crossing number calculated for variously knotted chains under modeled conditions of gel electrophoresis.

Complex knots on circular closed DNA molecules extracted from virus capsids have been observed by Atomic Force Microscopy after being adsorbed onto a surface. A characterization of the knot conformation according to the adsorption procedure is presented. The adsorption procedure has been previously characterized by measuring the scaling properties of a linear DNA molecule belonging to a different virus. The high resolution of the AFM images has allowed us to identify the chirality of some DNA crossings without any specific sample treatment. The high complexity of the knot population prevents us from identifying the knot types. The present results concern the determination of the three dimensional knot conformation from the two dimensional images.

Protein folds are considered as simple directed chains. Local smoothing of their coordinate sets (without chain passage) leads to a simple method to detect knots in open chains. A trefoil and figure-of-eight protein knot are described. Extension of the algorithm to incorporate hydrogen-bond cross-links allows protein pseudo-knots to be considered. The general connectivity of a protein chain is considered as a graph and a novel concept of topological accessibility described. These ideas are then applied to semi-random folds.

A mathematical knot is simply a closed curve in three-space. Classifying open knots, or knots that have not been closed, is a relatively unexplored area of knot theory. In this note, we report on our study of open random walks of varying length, creating a collection of open knots. Following the strategy of Millett, Dobay and Stasiak, an open knot is closed by connecting its two open endpoints to a third point, lying on a large sphere that encloses the random walk deeply within its interior. The resulting polygonal knot can be analyzed and its knot type determined, up to the indeterminacy of standard knot invariants, using the HOMFLY polynomial. With inany closure points uniformly distributed on the large sphere, a statistical distribution of knot types is created for each open knot. We use this method to continue the exploration of the knottedness of linear random walks and apply it also to the study of several protein chains. One new feature of this work is the use of an Eckert IV planar projection, preserving area, of the knotting distribution on the sphere to characterize the spatial properties of the distribution.

We compare here the scaling behaviour of the mean average crossing number 〈ACN〉 of equilateral random walks in linear and closed form with the corresponding scaling of natural protein structures. We have shown recently that the scaling of 〈ACN〉 of equilateral random walks of length n follows the relation and that a similar result holds for equilateral random polygons¹⁴. Furthermore, our earlier numerical studies indicated that when random polygons of length n are divided into individual knot types, the for each knot type can be described by a function of the form where a, b and c are constants depending on and n₀ is the minimal number of segments required to form ¹⁴. Here we analyze in addition natural protein structures and observe that the relation also describes accurately the scaling of 〈ACN〉 of protein backbones.

Basic folding features of a macromolecular chain can be conveyed by using a geometrical measure of chain entanglement, e.g., the mean over-crossing number (or "average crossing number") . In this work, we study the dependence of the configurationally-averaged value with the monomer number n, for a simple random-walk copolymer model involving two constant bond lengths b₁ and b₂. The results support the validity of the scaling , over the entire range of bond-length ratios z = b₁/(b₁ + b₂). The results serve as a benchmark for recognizing specific effects on polymer shape caused by chemical composition, or charging scheme, in realistic heteropolymers.

There is a striking qualitative similarity among the graphs of the relative probabilities of corresponding knot types across a wide range of random polygon models. In many cases one has theoretical results describing the asymptotic decay of these knot probabilities but, in the finite range, there is little theoretical knowledge and a variety of functional models have been used to fit the observed structures. In this paper we compare a selection of these models and study the extent to which each provides a successful fit for five distinct random knot models. One consequence of this study is that while such models are quite successful in this finite range, they do not provide the theoretically predicted asymptotic structure. A second result is the observed similarity between the global knot probabilities and those arising from small perturbations of three ideal knots.

In this paper, we extend results about the average crossing number of equilateral random walks and polygons to the average crossing number of the Gaussian random walks and polygons. We show that the asymptotical behavior of the ACN for the two models are very similar. More precisely, we show that the mean average crossing number 〈ACN〉 of Gaussian random walks and polygons of length n is of the form .

A physical interpretation of the rope simulated by the SONO algorithm is presented. Properties of the tight polygonal knots delivered by the algorithm are analyzed. An algorithm for bounding the ropelength of a smooth inscribed knot is shown. Two ways of calculating the ropelength of tight polygonal knots are compared. An analytical calculation performed for a model knot shows that an appropriately weighted average should provide a good estimation of the minimum ropelength for relatively small numbers of edges.

The ropelength of a space curve is usually defined as the quotient of its length by its thickness: the diameter of the largest embedded tube around the knot. This idea was extended to space polygons by Eric Rawdon, who gave a definition of ropelength in terms of doubly-critical self-distances (local minima or maxima of the distance function on pairs of points on the polygon) and a function of the turning angles of the polygon. A naive algorithm for finding the doubly-critical self-distances of an n-edge polygon involves comparing each pair of edges, and so takes O(n²) time. In this paper, we describe an improved algorithm, based on the notion of octrees, which runs in O(n log n) time. The speed of the ropelength computation controls the performance of ropelength-minimizing programs such as Rawdon and Piatek's TOROS. An implementation of our algorithm is freely available under the GNU Public License.

We discuss the linking probability of two random polygons, and evaluate the entropic force between them. We define the linking probability of a link L by the probability of a pair of random polygons of N nodes making the link L with their centers of mass separated by a distance R. Here we consider only such random polygons that have the trivial knot type. Through physical arguments we derive an analytic expression of the linking probability as a function of R, for the cases of the trivial and the Hopf links. Then we discuss the entropic force acting between such two random polygons forming the link L. Making use of the expression we evaluate the average size of random links, for the cases of a non-trivial link and the Hopf link. The average depends on the number of polygonal nodes, N, non-trivially.

We present computer simulations to examine probability distributions of gyration radius for the no-thickness closed polymers of N straight segments of equal length. We are particularly interested in the conditional distributions when the topology of the loop is quenched to be a certain knot . The dependence of probability distribution on length, N, as well as topological state are the primary parameters of interest. Our results confirm that the mean square average gyration radius for trivial knots scales with N in the same way as for self-avoiding walks, where the cross-over length to this "under-knotted" regime is the same as the characteristic length of random knotting, N₀. Probability distributions of gyration radii are somewhat more narrow for topologically restricted under-knotted loops compared to phantom loops, meaning knots are entropically more rigid than phantom polymers. We also found evidence that probability distributions approach a universal shape at N > N₀ for all simple knots.

We present a computer simulation study of the compact self-avoiding loops as regards their length and topological state. We use a Hamiltonian closed path on the cubic-shaped segment of a 3D cubic lattice as a model of a compact polymer. The importance of ergodic sampling of all loops is emphasized. We first look at the effect of global topological constraint on the local fractal geometry of a typical loop. We find that even short pieces of a compact trivial knot, or some other under-knotted loop, are somewhat crumpled compared to a topology-blind average over all loops. We further attempt to examine whether knots are localized or de-localized along the chain when the chain is compact. For this, we perform computational decimation and chain-coarsening, and look at the "renormalization trajectories" in the space of knots frequencies. Although not completely conclusive, our results are not inconsistent with the idea that knots become de-localized when the polymer is compact.

We discuss the competition between the topological effect and the excluded volume effect on the average size of ring polymers with a fixed knot, reviewing some simulation results of Ref.¹⁹. Under a topological constraint, the average size of ring polymers can be much larger than that of no topological constraint. However, the effective expansion depends strongly on the excluded-volume parameter. We also discuss the radial distribution functions of segments of ring polymers with fixed knots. The numerical results of the radial distributions suggest that a topological constraint on ring polymers effectively leads to an entropic repulsion among polymer segments.

We investigate statistical properties of random lattice knots, the topology of which is determined by algebraic topological invariants. Representing the Jones-Kauffman polynomial invariant of a random knot diagram by a partition function of a Potts model with random coupling constants, we study the probability distribution of different topological types of random lattice knots. We are interested in the probability of the "knottedness" (the "knot complexity") of densely packed knots, which we measure via the degree of their Jones-Kauffman polynomial. In particular, we find the mean complexity of "daughter knots," obtained by cutting off a part of a trivial (i.e. unknotted) "parent" densely packed knot and closing up the "open tails" (the loose ends of the cut strands). We present arguments supporting the conjecture that the knot complexity n* of a daughter knot of an unknotted parent one grows as , where N is the total number of vertices on the lattice. This result gives a strong support for the conjectured "crumpled globule" structure of collapsed unknotted closed polymer chains, in which the polymer forms a system of densely packed folds, mutually separated in a broad range of scales.

In this paper, the problem of generating large random knot projections is explored. Regular knot projections without the usual over and under information at each crossing the can be viewed as 4-regular planar graphs. Two methods are introduced that generate 4-regular planar graphs that can be viewed as projections of prime knots. Various questions that arise along the way are explored.

This paper gives mathematical models for flat knotted ribbons, and makes specific conjectures for the least length of ribbon (for a given width) needed to tie the trefoil knot and the figure eight knot.

It is known that every nontrivial knot has at least two quadrisecants. In this paper we illustrate quadrisecants for several configurations of the knots of crossing number up to five. Given a knot, we mark each intersection point of each of its quadrisecants; replacing the subarcs between marked points by straight segments gives a modified knot called the quadrisecant approximation. We conjecture that this always has the same knot type as the original.

The paper deals with the definition and computation of the writhing number of an arbitrary fragment of a space curve. The approach is based on closing the tangent indicatrix with a geodesic. A relationship connecting the writhe with the Gauß integral over the open curve is studied. Single and double helical shapes are presented as examples.

Motivated by previous work on elastic rods with self-contact, involving the concept of the global radius of curvature for curves (as defined by Gonzalez and Maddocks), we define the global radius of curvature ∆[X] for a wide class of parametric surfaces. It turns out that a positive lower bound ∆[X] ≥ θ > 0 provides, naively speaking, the surface with a thickness of magnitude θ it serves as an excluded volume constraint for X, prevents self-intersections, and implies that the image of X is an embedded C¹-manifold with a Lipschitz continuous normal. Taking into account possible applications to variational problems for embedded objects, we also obtain a convergence and a compactness result for such thick surfaces.

The main object of this note is to introduce and explain the crucial notions and results. Thus, we defer almost all proofs to our forthcoming paper.²²

We describe several configurations of clasped ropes which are balanced and thus critical for the Gehring ropelength problem.

In this paper we describe a recursive technique that can be used to investigate a number of hyperbolic invariants of certain classes of 2-bridge knots. We explicitly compute the representation and character varieties, the trace and cusp fields, and A-polynomials for these knots. Using these computations we obtain information about these knots' commensurability classes. Experimental results lead naturally to two conjectures regarding trace fields and commensurability classes of 2-bridge knots. We conclude with a discussion of a result of Hodgson which generalizes other results of this paper.

We show how some issues of enumeration in knot theory are related to statistical models on random lattices. The insight it gives us leads to conjectures on asymptotic counting as the number of crossings becomes large. We describe some numerical work to test these conjectures. We briefly outline a generalization to virtual knot theory.

The following sections are included:

• SERIES ON KNOTS AND EVERYTHING