Ideal Knots

The following sections are included:

• Preface

• CONTENTS

We present here the concept of ideal geometric representations of knots which are defined as minimal length trajectories of uniform diameter tubes forming a given type of knot. We show that ideal geometric representations of knots show interesting relations between each other and allow us to predict certain average properties of randomly distorted knotted polymers. Some relations between the behaviour of real physical knots and idealised representations of knots which minimise or maximize certain properties were previously observed and discussed in Ref. 1-5.

A particular version of the knot tightening algorithm is described. It is shown that the algorithm is able to remove empty loops and nugatory crossing leading to the simplification of the conformation of any knot. The problem of finding the ground state conformation of knots is discussed. Results of tightening various types of knots are presented and analysed.

Simulated annealing is a powerful method for finding global minima. The application of annealing to the search for ideal knots and links is explored, and some optimisations and problems discussed.

Energy functions on knots are continuous and scale-invariant functions defined from knot conformations into non-negative real numbers. The infimum of an energy function is a knot invariant which defines (not necessarily unique) “canonical conformations” of knots in three space. In this paper we examine strategies of how to find these canonical conformations and compute or measure their minimal energy. Furthermore, we discuss properties that energy functions should have if one wants to compute canonical conformations of knots with minimal energy. Two types of energies are discussed in more detail, the first type consists of energies of C² or C¹ knots defined using the concept of thickness of a knot; the second type is a polygonal energy which should be well suited to numerical computation since it has all the properties discussed earlier.

We discuss the writhe of linked and knotted simple closed curves embedded in the simple cubic lattice, Z³. We show that the writhe of a simple closed curve in Z³ can be computed as the average of its linking numbers with certain pushoffs, and use this result to establish a lower bound on the rate of increase of the mean absolute writhe. We present Monte Carlo results on the distribution of writhe for particular knot types, and compare the mean values with values for ideal knots. Similar results are presented for links and we show that the mean writhe of (2, 2k) torus links increases linearly with crossing number.

How many unit edges and right angles are needed to construct a knot of a given type in the cubic lattice. The unknot can be made using 4 edges and 4 right angles, since the girth of the cubic lattice is 4, and the curvature of a lattice polygon must be at least 2π (define curvature of a piecewise linear curve as the sum over the excluded angles between successive line segments). It is also known that the trefoil can be constructed in the cubic lattice using 24 edges (and not with any fewer), and that it will have at least 12 right angles. In this chapter I review the properties of minimal lattice knots, concentrating of the minimum length and minimum curvature for given knot types.

The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or “geometric” knots, while paths correspond to isotopies which preserve the geometric structure of these knots. Two cases are considered: (i) the space of polygons with varying edge length, and (ii) the space of equilateral polygons with unit-length edges. In each case, the spaces are explored via a Monte Carlo search to estimate the distinct knot types represented. Preliminary results of these searches are presented. Additionally, this data is analyzed to determine the smallest number of edges necessary to realize each knot type with nine or fewer crossings as a polygon, i.e. its “minimal stick number.”

The idea of maximally inflated tube representation of a knot is employed to examine the question of knot entropy, which essentially reduces to how many “conformations” are there corresponding to a given knot topological type. Simple scaling arguments are given, bringing the entire seemingly intractable problem of knot entropy into the realm of simple conventional methods.

The thickness of a unit length C² knot is the radius of the thickest “rope” one can place about a knot at the instant that the “rope” self-intersects. Thickness is difficult to compute for all but a few examples. To use computers, a polygonal version of thickness must be defined. The most natural definition does not correctly approximate thickness so a different polygonal version is necessary. This paper contains a definition of a continuous polygonal thickness which correctly approximates smooth thickness. Results on approximation and continuity are stated and examples given of thickness approximations.

Several definitions have been proposed for the “energy” of a knot. The intuitive goal is to define a number u(K) that somehow measures how “tangled” or “crumpled” a knot K is. Typically, one starts with the idea that a small piece of the knot somehow repels other pieces, and then adds up the contributions from all the pieces. From a purely mathematical standpoint, one may hope to define new knot–type invariants, e.g by considering the minimum of u(K) as K ranges over all the knots of a given knot–type. We also are motivated by the desire to understand and predict how knot–type affects the behavior of physically real knots, in particular DNA loops in gel electrophoresis or random knotting experiments. Despite the physical naiveté of recently studied knot energies, there now is enough laboratory data on relative gel velocity, along with computer calculations of idealized knot energies, to justify the assertion that knot energies can predict relative knot behavior in physical systems. The relationships between random knot frequencies and either gel velocities or knot energies is less clear at this time.

The article describes a tool for simplification and analysis of tangled configurations of mathematical knots. The proposed method addresses optimization issues common in energy-based approaches to knot classification. In this class of methods, an initially tangled elastic rope is “charged” with an electrostatic-like field which causes it to self repel, prompting it to evolve into a mechanically stable configuration. This configuration is believed to be characteristic for its knot type. We propose a physically-based model to implicitly guard against isotopy violation during such evolution and suggest that a robust stochastic optimization procedure, simulated annealing, be used for the purpose of identifying the globally optimal solution. Because neither of these techniques depends on the properties of the energy function being optimized, our method is of general applicability, even though we applied it to a specific potential here. The method has successfully analyzed several complex tangles and is applicable to simplifying a large class of knots and links. Our work also shows that energy-based techniques will not necessarily terminate in a unique configuration, thus we empirically refute a prior conjecture that one of the commonly used energy functions (Simon's) is unimodal. Based on these results we also compare techniques that rely on geometric energy optimization to conventional algebraic methods with regards to their classification power.

Please refer to full text.

The following sections are included:

• Introduction

• The general theory of relaxation

• Conservation of field helicity

• Relaxation of knotted fields

• Relaxation of strongly twisted tubes

• References

Bistable chemically reacting media can segregate into domains of two stable phases that differ in their chemical composition and are separated by chemical fronts. For activator-inhibitor kinetics, when the diffusion coefficient of the inhibitor is greater than that of the activator, stable localized chemical patterns can form. These localized structures arise from the interaction between chemical fronts as a consequence of the fast inhibitor diffusion. In three dimensions one can build linked and knotted chemical dissipative structures whose stability is a consequence of their topology. In the general case, the evolution equations that govern the pattern formation are not of gradient type but the stable chemical patterns assume nearly ideal forms.

In this paper we review classical and new results in topological fluid mechanics based on applications of first principles of ideal fluid mechanics and knot theory to vortex and magnetic knots. After some brief historical remarks on the first original contributions to topological fluid mechanics, we review basic concepts of topological fluid mechanics and local actions of fluid flows. We review some classical, but little known, results of J.J. Thomson on vortex links, and discuss Kelvin's conjecture on vortex knots. In the context of the localized induction approximation for vortex motion, we present new results on existence and stability of vortex filaments in the shape of torus knots. We also discuss new results on inflexional magnetic knots and possible relaxation to minimal braids. These results have potentially important applications in disciplines such as astrophysics and fusion plasma physics.

We describe a first principles Hamiltonian approach, where knots appear as solitonic solutions to the pertinent nonlinear equations of motion. This makes it possible to predict all properties of knotted configurations in terms of fundamental data that are only characteristics of the underlying physical environment. In particular, no knot-specific parameters appear in this approach.

A knot energy functional is a real valued functional on the space of knots that blows up if a knot has self-intersections. We consider the regularization of modified electrostatic energy of charged knots. We study several kinds of knot energy functionals and consider whether there exists an energy minimizer in each knot type, which is characterized as an embedding that attains the minimum value of the knot energy functional within its ambient isotopy class.

There has been recent interest in knot energies among mathematicians and natural scientists. When discretized, such energies can lead to effective algorithms for recognizing when two curves represent the same knot. These energies may also help model physical systems, such as long protein chains or DNA knots, subject to van der Waals interactions. Knot energies often are normalized to be scale-invariant; some important energies are also invariant under Möbius transformations of space. We describe computer experiments with such Möbius-invariant knot energies. We also discuss ways of extending these to energies for higher-dimensional submanifolds. The Appendix gives a table of computed Möbius-energy-minimizing knots and links through eight crossings. (This article is an updated version of our report¹ in Geometric Topology.)

An harmonic knot is a knot in three-dimensional space that can be expressed parametrically in the form (x(t), y(t), z(t)) where each coordinate function is a trigonometric polynomial. Since every knot is ambient isotopic to an harmonic knot, we are able to define the harmonic index of a knot as the smallest degree of any harmonic knot in the knot class. The harmonic index of a knot is related to its superbridge index and crossing number. In particular, there is a bound on harmonic index computed from the crossing number and vice versa. This bound allows us to compute the superbridge index of the figure-eight knot and the granny knot and to show that only finitely many knot types occur with any given harmonic index. We provide explicit harmonic parametrization of the trefoil knot, the figure-eight knot, the (2, 5) and the (3, 4) torus knots, and the granny knot.

This paper introduces the concept of Fourier knot. A Fourier knot is a knot that is represented by a parametrized curve in three dimensional space such that the three coordinate functions of the curve are each finite Fourier series in the parameter. That is, the knot can be regarded as the result of independent vibrations in each of the coordinate directions with each of these vibrations being a linear combination of a finite number of pure frequencies.

The following sections are included:

• Symmetric knots

• Polynomial invariants of links

• Periodic links and the Jones polynomial

• Periodic links and the generalized Jones polynomials

• r^q-periodic links and Vassiliev invariants

• Lissajous knots and billiard knots

• Applications and Speculations

• References