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Knots at Leeds


Textile Structure

New areas of application of textile fibrous assemblies for automotive, aerospace and medical materials, smart and sensory materials, protective clothing, wearable electronics, pervasive computing, and other modern areas require an application-specific engineering approach. Despite growing understanding of a significant dependence between structural characteristics and physical properties of fibrous assemblies, and the latest developments in computer technology, mainstream research in textiles is still focused on relationships between internal geometry and physical properties. This can be explained by the lack of a universal numerical parameter that would be able to characterise the structure of fibrous assemblies.

Structural characteristics of textiles depend on the mutual position of constituting elements, i.e. fibres and threads. Textile structures are specific because many of them can be defined by a unit cell repeated in a periodic manner across the fabric so that they can be considered as a particular case of periodic interlaced structures in 3-dimensional space. On the other hand, there are textiles where structural elements are randomly arranged into a string-like or sheet material. Significant differences in the geometrical characteristics have led to the fundamental differences in the methods of description of their structural and physical properties despite the fact that they all have the same parent product which is fibre.

The overall vision for the project is that using more general theory - the topological theory of knots, links, braids, and tangles - it will be possible to find a common method that could be suitable for the description and classification of fibrous structures from different classes. These mathematical objects can be naturally associated with physical objects such as textile fibrous assemblies because both types essentially are spatial thread-based objects. The differences between the two are in fine details of the properties of the threads which must be assumed to be infinitely thin, frictionless and continuously deformable without breaking or passing through each other. Under these assumptions standard mathematical definitions can be used and indeed have been used in application to textile structures. Until now, however, there have been very few attempts to use topological methods for describing the structure-dependent properties of textiles. Some limited success has been achieved in the application of topological invariants for the classification of textiles and for identification of non-interlaced layers and threads, and closed loops which are not true elements of textiles.

There are physical and mechanical properties of textiles which depend on both structure and geometry. Generating an invariant which is sensitive to such fine features presents a significant difficulty from the theoretical point of view and requires substantial efforts in fundamental research. To our knowledge, invariants with such properties have never been directly considered in knot theory. This project is the first attempt to call upon the joint expertise and knowledge of specialists in topology and modern numerical methods together with a substantial involvement of textile scientists.

Molecular and biological knots

Knots are prevalent at molecular length scales, particularly in larger polymer molecules such as synthetic and biological macromolecules. As the length/size increases, the probability that the backbone of the molecule is intertwined or even knotted increases, the exact probability being determined by the degree of compactness which in turn depends on the intramolecular interactions. If a molecule is a closed loop it can become either knotted with itself or catenated with another closed loop molecule. These generalities of behaviour apply to synthetic organic constructs as well as biological molecules.

This project aims to understand some general behaviour of molecular knots and relate it to knot theories and other themes in LINK, such as textiles and braids. A deeper understanding of molecular knots requires a multidisciplinary approach including synthesis, characterisation and modelling. It groups together expertise and interest at Leeds to study the properties and interactions of molecules with propensity to knot or entangle, from synthesised molecule knots to natural biological molecules such as polypeptides and nucleic acids (e.g. DNA).

At the smaller end of the size scale, supramolecular species can be induced to assembly into knots or other topologically complex motifs such as catenanes or rotaxanes through directed or template self-assembly. The MH group will synthesise target knots through directed self-assembly approaches and work with the SH group to study the motion of these molecules through molecular dynamics simulation. Knotted structures can be elucidated through X-ray crystallography and advanced NMR techniques available at Leeds, and some will be large enough for AFM visualisation.

The atomic force microscope (AFM) is a high resolution microscope that can visualise the structure of single molecules, such as the DNA helix, by feeling with a nano-sized stylus. It can be operated in liquid environments to visualise molecular structures in situ and thereby follow dynamic changes, either natural fluctuations or stimulated alterations. By 2014, the NT group will have a fast scanning AFM that can directly follow processes involved with molecular knots. AFM will visualise knot behaviour and structure at high resolution and outcomes will be compared with theoretical and atomistic models to deepen our understanding of biological molecules that create knots, particularly those in proteins and nucleic acids.

The topological properties of biomolecules have important consequences for biological control and function. While knots can only be rigorously defined for closed chains, the definition can be extended to open chains by projecting the ends to infinity using widely accepted procedures. Knots have been found in some proteins but their rarity suggests evolution has pushed towards removal of knots. Knots may stabilise folded states but introduce frustration in folding landscapes and complicate the folding process, and their possible role in proteins' function is unknown. If evolution discouraged (without completely preventing) the presence of knots, then knots should occur more frequently in random polypeptides.

Intrinsically disordered proteins (IDPs) are polypeptides that naturally occur within biology and are important for biological signalling but do not fold up into a 3D ordered structure like other proteins, instead fluctuating between different states. A timely question is whether knots occur more frequently in IDPs than in proteins of the same length whose function is connected to their ability to fold to a precise structure. Fast scan AFM in situ is particularly suited to looking at IDPs, because they are unstructured and therefore difficult to study with high resolution diffractive techniques (e.g. crystallography). AFM will capture the dynamics of IDPs of various sizes and be compared with modelling to learn about the probability of knotting in these unstructured signalling molecules.

Structure and properties of industrial compounds

Within a polymeric fluid, the chain-like molecules are arranged randomly, and are in a constant state of agitated motion due to their heat energy. Some well-established models exist, in which the statistical properties of the polymers\' space-curves govern the large-scale properties of the fluid, such as its viscosity, elasticity and miscibility.

As a polymeric fluid flows, the chains cannot pass through each other, so their topology is of crucial importance in controlling their shapes, and hence the fluid\'s properties and behaviour. Nevertheless, the topology of a collection of polymers and its connection to the fluid\'s large-scale mechanics has not been studied in detail. Instead the topological properties are conventionally modelled in terms of a density of ill-defined \'entanglements\', that are assumed to behave like chemical bonds.

Using existing simulation tools, a careful investigation of the effects of specific topological arrangements of polymers will allow us to discover which topological features most strongly influence large-scale behaviour. This will allow better polymeric models to be developed, and better control over industrial and biological fluids.
Colloids, such as emulsions used in foods and household goods, can also form topologically non-trivial networks. The difference is that their thread-like structures can break and re-form as the material flows, altering their physical behaviour. Their models inherit many features from polymeric models, but provide a useful counterexample in which the importance of the topology can be compared with the influence of other physical and geometrical features.

Such an investigation requires a collaboration between experts in polymer physics and experts in knot theory, and will share features and knowledge with the investigation into textiles.

Topological Quantum Computation

A goal in topological quantum computation (TQC) is to identify quantum states of matter described by knots, links and higher dimensional generalisations. Some of these states support braidable anyonic excitations. The knotted property is immune to noise, and hence key for quantum technologies. As such TQC leads us to study many aspects of the nature of knots. Here we describe an indicative subset.

Anyons in Chern-Simons (CS) field theory have quantum mechanical amplitudes given by Jones polynomials. But CS theories describe only a particular class of anyons. We seek new invariants associated to other types of anyons. Indeed we need to extend the concept of knots to describe physical anyons. This includes geometric and dynamical extensions. For example, Abelian anyons in CS theories may be accompanied by a Maxwell term. The anyon evolutions corresponding to the CS part are given by their worldline Jones polynomials. It is an open question how to accommodate the Maxwell term. Such extensions are also relevant in polymer and DNA physics. Generalised loop braiding algebras are intriguing. Based on a bulk-boundary correspondence, they still appear to lead back to anyonic/point braiding - suggesting no new physics in 3d topological systems. But we can bypass this correspondence using boundary theories beyond those definable in 2d. Such theories could support trans-anyonic statistics. Here a systematic characterisation of loop statistics will be developed, and applied to 3d systems.

Realisable anyonic models are sometimes not complex enough to support universal quantum computation just by unitary braiding. Measurement is a computational primitive that, combined with non-universal unitaries, can result in universal quantum computation. We aim to make anyons such as parafermions and Majorana fermions universal via charge measurements. It may also be possible to construct quantum gates from Fibonacci anyons, thus reducing the current requirement of many braiding operations for a single CNOT gate.

For application to the statistical physics of string-like quantum systems, we aim to understand the behaviour of an anyonic quantum walker that moves by braiding around stationary anyons. The asymptotic behaviour involves evaluating Jones polynomials with many crossings. We already determined asymptotics for the case of Abelian anyons and SU(2) level 2 non-Abelian anyons. We also need to consider the effect of Anderson localisation of the walker when the topological background is randomised. We already determined that Abelian anyons localise while SU(2) level 2 non-Abelian anyons do not. Determining the diffusion properties can help identify anyons in real physical devices from their transport properties.

Knots invariants

Amongst the algebraists in Leeds we have a unique combination of expertise in three areas on mathematics relevant to knot theory: diagram algebras, cluster algebras, and homological algebra. Diagram algebras are defined graphically, based on relations arising from knot diagrams. Cluster algebras were defined in order to study bases of quantum groups, which can be used to study knot invariants, and Skein relations between curves on surfaces are important in finding bases for cluster algebras. Also, recent research conducted at Leeds has connected cluster algebras to generators of braid groups. Homological algebra was connected to knot theory by Khovanov, who used categorification, the art of replacing a simple mathematical object by a collection of more complicated and structured objects, to produce a homology theory which recovers, but is much more powerful than, the Jones polynomial.
An important recent development is the construction by Webster of knot invariants by generalizing the Khovanov-Lauda-Rouquier categorification of quantum groups. This generalizes Khovanov\'s theory, which is associated to the defining representation of sl(2), to arbitrary representations of simple Lie algebras.

We outline two specific projects we would undertake in this area. The first involves making specific calculations of Webster\'s invariants. So far, the Webster invariants are not known in general even in the simplest case of the unknot (i.e., a circle!). The problem is purely one of computation in noncommutative homological algebra, which we are well-suited to tackle in Leeds. We would first solve the unknot case, then consider tangles arising from braids on small numbers of strands and investigate general techniques. The second problem is the conjectural relationship between Mazorchuk-Stroppel-Sussan homology and Khovanov-Rozansky homology. We would compare these two theories using Khovanov\'s reformulation of Khovanov-Rozansky homology in terms of Soergel bimodules, which avoids matrix factorizations and gives an explicit connection to the Lie theory.

Magnetic knots

It is widely appreciated that knots in magnetic plasmas should be regarded as framed knots. Progress in studying magnetic knots has been hampered by the numerical difficulties of simulating fully three-dimensional fields with discontinuities. Researchers in Leeds are well-placed to overcome these difficulties: we have substantial experience studying the Faddeev-Skyrme model, a more mathematical model which nevertheless has many features in common with magnetic plasmas. By exploiting these common features we propose to set up numerical simulations which produce energy-minimising magnetic knots. Moreover, by applying techniques successfully already developed we will develop an effective theory of magnetic knots based on Kirchhoff\'s energy. This would have two positive consequences: first, it would enable predictions to be made for magnetic knots which can then be tested by direct simulation; and second, linking Kirchhoff\'s energy to the magnetic energy may yield a topological lower bound on Kirchhoff\'s energy, as it is already known that the magnetic energy is bounded from below by a topological invariant.

Chaotic mixing by topological braiding

The complexity inherent in repeatedly braiding worldlines can be harnessed to mix fluids. Essentially this is stirring braids into a fluid, where chaotic motion guarantees that fluid elements stretch at exponential rate. This lower bound on the stretching of material lines is now well-understood, but this is a purely topological view. A different analysis, using ergodic theory of dynamical systems, of the same model systems shows that correlations decay at algebraic, not exponential, rate, due to the effect of boundary behaviours. This project aims to bridge the gap between the topological and measure-theoretic dynamics viewpoints, quantifying how far the mixing effect of the braiding extends into the whole fluid domain, to produce a set of design criteria for a wide class of fluid mixing device. The approach is a mixture of analysis, using tools such as Young Towers from ergodic theory and Thurston-Nielsen classification from topological dynamics, and numerical work, computing topological and measure-theoretic entropy, and related quantities.

Spatial reasoning with knots

Qualitative spatial reasoning and representation deals with commonplace spatial relationships such as might be used in tasks including lifting one object to retrieve another or formulating plans involving geographical regions that might overlap or surround others. The formalization of these relationships (frequently topological rather than geometric) has only recently been considered in the context of knots, where a task may involve passing threads or objects through holes in rigid objects or through loops created by manipulating threads. The long-term aim of constructing a robot that could tie and untie knots is well beyond the scope of this project but a significant step towards this is the development of a logical theory of how physical knots can be manipulated, a problem distinct from manipulating knot diagrams. The logical theory will need to combine aspects of existing qualitative calculi and formal ontology (such as the Region Connection Calculus developed extensively at Leeds) with mathematical techniques from knot theory.