# Workshop Programme

## for period 3 - 7 December 2012

### Quantised Flux in Tightly Knotted and Linked Systems

3 - 7 December 2012

Timetable

 Monday 3 December 09:00-10:15 Registration 10:15-10:30 Opening remarks by Deputy Director, Christie Marr, and Keith Moffatt Session: Topological evolution in quantum fluids Chair: Keith Moffatt 10:30-11:10 Barenghi, CF (Newcastle University) The properties of quantum turbulence: the homogeneous isotropic case Sem 1 Quantum mechanics constrains the rotational motion of superfluid helium to discrete, vortex filaments of fixed circulation and atomic thickness (quantum vortices). A state of "quantum turbulence" can be easily created by stirring the liquid helium thermally or mechanically. In this talk I shall review recent experiments and numerical calculations which have revealed remarkable similarities between this form of turbulence and turbulence in ordinary fluids. Classical behaviour (such as the celebrated Kolmogorov energy spectrum) seems to arise from the coherence of many quanta of elementary circulation. 11:10-11:30 Kerr, RM (Warwick University) Quantum and classical turbulence: Alike or different? Sem 1 The question of how alike and different quantum and classical turbulence are will be addressed using simulations of anti-parallel vortex reconnection. The equations solved are: For quantum fluids the Gross-Pitaevskii equation and for classical turbulence the incompressible Navier-Stokes equation. For the two cases the initial attraction of the vortices, before the first reconnection, are quite similar. And for both, the final states are composed of a stack of vortex rings from which a -5/3 kinetic energy spectrum appears. However, almost everything in between is different, starting with the differences in the underlying physics of the circulation during reconnection. This presentation will describe these differences. 11:50-12:10 Sherwin, L (Newcastle University) Thermally and mechanically driven quantum turbulence in helium II Sem 1 In most experiments with superfluid helium, turbulence is generated thermally (by applying a heat flux, as in thermal counterflow) or mechanically (by stirring the liquid). By modeling the superfluid vortex lines as reconnecting space curves with fixed circulation, and the driving normal fluid as a uniform flow (for thermal counterflow) and a synthetic turbulent flow (for mechanically driven turbulence), we determine the difference between thermally and mechanically driven quantum turbulence. We find that in mechanically driven turbulence, the energy is concentrated at the large scales, the spectrum obeys Kolmogorov scaling, vortex lines have large curvature, and the presence of coherent vortex structures induces vortex reconnections at small angles. On the contrary, in thermally driven turbulence, the energy is concentrated at the mesoscales, the curvature is smaller, the vorticity field is featureless, and reconnections occur at larger angles. Our results suggest a method t o experimentally detect the presence of superfluid vortex bundles. 12:10-12:30 Allen, J (Newcastle University) Generating and Classifying Turbulence in Bose-Einstein condensates Sem 1 Vortices are a hallmark signature of a turbulent flow. Quantum vortices differ from their classical counterparts because of the quantization of circulation in superfluid flow. This means that the rotational motion of a superfluid is constrained to discrete vortices which all have the same core structure. Turbulence in superfluid Helium has been the subject of many recent experimental and theoretical investigations recently reviewed by Skrbek and Sreenivasan [1]. Recently, experimentalists have been able to visualise individual vortex lines and reconnection events using tracer particles[2]. Weakly interacting Bose-Einstein condensates present a unique opportunity to resolve the structure of vortices and in turn study the dynamics of a vortex tangle (as has recently been created in an atomic cloud[3]). We investigate ways of generating turbulence in atomic systems by numerically stirring the condensate using a Gaussian 'spoon' (analogous to a laser beam in the experiments), and study the isotrophy of the resulting vortex tangle depending on when the path the spoon stirs is circular or random. We model the system using the Gross-Pitaevskii Equation. [1] L. Skrbek and K.R. Sreenivasan, PoF 24, 011301 (2012) [2] G.P. Bewley et al. PNAS 105, 13707 (2008). [3] E.A.L. Henn et al. PRL 103, 04301 (2009). 11:30-11:50 Morning Coffee 12:30-13:30 Lunch at Wolfson Court Session: Topological gauge theories and particle physics Chair: Roman Buniy 14:00-14:40 Kauffman, LH (University of Illinois at Chicago) Topological Models for Elementary Particles Sem 1 The talk will be a survey of topological models for elementary particles including the work of Lord Kelvin, Herbert Jehle, Thomas Kephart, Jack Avrin, Sundance Bilson-Thompson and recent work of the speaker with Sundance Bilson-Thompson and Jonathan Hackett and work of the speaker on the Fibonacci model in quantum information theory. Lord Kelvin suggested that atoms (the elementary particles of his time) are knotted vortices in the luminiferous aether. Jehle (much later on) suggested that elemenary particles should be quantized knotted electromagnetic flux. Kephart and Buiny suggest that closed loops of gluon field can be knotted particles -- knotted glueballs. Avrin notes that a three half-twisted Mobius band could be like a proton composed of three quarks mutually bound. Sundance Bilson-Thompson has a theory of framed three-braids that is a topological version of preons. In this theory we can think of particles as topological defects in networks of surfaces and some properties of embedded surfaces may sort out the matter. In the Fibonacci model for topological quantum computing, the system is generated by a braided anyonic abstract particle P that interacts with itself to produce itself (PP -----> P) or iteracts with itself to produce a neutral particle (PP ------> 1). The elementary particle P of the Fibonacci model is a structure that can be seen as a logical particle, underlying all the mathematical structures that we know. Since this talk surveys such a range of ideas, it will be up to the speaker to find a way to summarize these diseparate views at the time of the talk. 14:40-15:00 Kephart, TW (Vanderbilt University) The spectrum of tightly knotted flux tubes in QCD Sem 1 15:20-16:00 Duff, M (Imperial College London) Quantized black hole charges and the Freudenthal dual Sem 1 It is well-known that the quantized charges x of 4D black holes may be assigned to elements of an integral Freudenthal triple system (FTS). The FTS is equipped with a quartic form q(x) whose square root yields the lowest order black hole entropy. We show that a subset of these black holes, for which q(x) is necessarily a perfect square, admit a Freudenthal dual'' with integer charges ~x, for which ~~x=-x and q(~x)=q(x). [1] ''Black holes admitting a Freudenthal dual'', L. Borsten, D. Dahanayake, M.J. Duff, W. Rubens, Phys.Rev. D80 (2009) 026003 e-Print: arXiv:0903.5517 [hep-th] [2] '' Freudenthal dual invariant Lagrangians'', L. Borsten, M.J. Duff, S. Ferrara ana A, Marrani (to appear). 16:00-16:20 Päs, H (TU Dortmund) Knotted strings and leptonic flavor structure Sem 1 Tight knots and links arising in the infrared limit of string theories may provide an interesting alternative to flavor symmetries for explaining the observed flavor patterns in the leptonic sector. As an example we consider a type I seesaw model where the Majorana mass structure is based on the discrete length spectrum of tight knots and links. It is shown that such a model is able to provide an excellent fit to current neutrino data and that it predicts a normal neutrino mass hierarchy as well as a small mixing angle $\theta_{13}$. 16:20-16:40 Dennis, M (University of Bristol) Designing fibred knots in optical fields Sem 1 17:00-18:00 Berry, M (H H Wills Physics Laboratory, University of Bristol) Rothschild lecture: Superoscillations and weak measurement Sem 1 Band-limited functions can oscillate arbitrarily faster than their fastest Fourier component over arbitrarily long intervals. Where such ‘superoscillations’occur, functions are exponentially weak. In typical monochromatic optical fields, substantial fractions of the domain (one-third in two dimensions) are superoscillatory. Superoscillations have implications for signal processing, and raise the possibility of sub-wavelength resolution microscopy without evanescent waves. In quantum mechanics, superoscillations correspond to weak measurements, suggesting ‘weak values’ of observables (e.g photon momenta) far outside the range represented in the quantum state. A weak measurement of neutrino speed could lead to a superluminal result without violating causality, but the effect is too small to explain the speed recently claimed in a recent (and now-discredited) experminent. 15:00-15:20 Afternoon Tea 18:00-18:30 Welcome Wine Reception
 Tuesday 4 December Session: Topological evolution in gauge theories and cosmology Chair: Tanmay Vachaspati 08:50-09:30 Achucarro, A (Leiden University) Cosmic strings and other flux tubes Sem 1 An introduction to cosmic strings and superstrings including a status report of the observational bounds. 09:30-09:50 Shellard, P (University of Cambridge) Cosmic Strings and our Cosmos? Sem 1 The talk will be about observational constraints on cosmic strings, notably those from searching for evidence of their signatures in the cosmic microwave sky. I will review current constraints on cosmic strings and future prospects from experiments like Planck. 09:50-10:30 Hindmarsh, M (Helsinki Institute of Physics/University of Sussex) Cosmic strings - relativistic vortices in the early universe Sem 1 In many models of the very early universe, a tangle of relativistic vortices - cosmic strings - are spontaneously formed at a phase transition. I will briefly review the models and observational constraints, compare and contrast cosmic strings with superconductor flux tubes and superfluid vortices, and describe the latest large-scale simulations of their formation and decay. 10:50-11:10 Pogosian, L (Simon Fraser University) Scaling configurations of cosmic superstring networks and their cosmological implications Sem 1 Cosmic superstring networks contain strings of multiple tensions and Y-junctions. Depending on the magnitude of the fundamental string coupling, the stress-energy power spectrum of the scaling network is dominated either by populous light F strings, or rare heavy D strings. It may be possible to distinguish between these two regimes with future observations of CMB polarization. 11:10-11:30 Martins, C (CAUP) Scaling properties of cosmic superstring networks Sem 1 I will use a combination of state-of-the-art numerical simulations and analytic modeling to characterize the scaling properties of cosmic superstring networks. Particular attention will be given to the role of extra degrees of freedom in the evolution of these networks. Compared to the 'plain vanilla' case of Goto-Nambu strings, three such extensions play important but distinct roles in the network dynamics: the presence of charges/currents on the string worldsheet, the existence of junctions, and the possibility of a hierarchy of string tensions. I will also comment on insights gained from studying simpler defect networks, including Goto-Nambu strings themselves, domain walls and semilocal strings. 11:30-11:50 Steer, D (Université Paris 7 - Denis-Diderot) String networks with junctions: evolution and kinks Sem 1 I will discuss a number of properties of string networks with junctions, focusing particularly on their evolution and the number of kinks in such networks. These properties are crucial in order to determine the observational consequences of networks with junctions, for instance in Gravitational Waves and CMB polarization. 11:50-12:10 Sakellariadou, M (King's College London) Constraints on cosmic superstring formation and their decay mechanism Sem 1 I will discuss some aspects of cosmic superstrings, related to their formation, nature, evolution and decay mechanisms. In particular, I will discuss the successful and theoretically consistent brane inflation models leading to cosmic superstrings, the dynamics of junctions and the channels of decay mechanisms of cosmic superstrings, which are very important for cosmological/observational consequences. 12:10-12:30 Urrestilla, J (University of the Basque Country) Cosmic strings, effective number of neutrinos, gravity waves and the CMB Sem 1 We will report about recent work concerning cosmic defects, gravity waves and the CMB. On one hand, we simulate a scaling network of cosmic defects, and predict the background gravity wave emission coming from them. On the other, we study degeneracies in the CMB data between cosmic strings and gravity waves. When one tries to fit the CMB data with the effective number of neutrinos as a free parameter, the data prefers a higher number effective neutrinos. There have been works suggesting that the extra relativistic signal could come from gravity waves, and those waves could be created by cosmic defects. We perform a full analysis of the system taking into account the signal that such a network would create, its effect into the effective number of neutrinos, and how those parameters interplay in trying to fit different probes of CMB and other cosmological data. 10:30-10:50 Morning Coffee 12:30-13:30 Lunch at Wolfson Court Session: Topology of condensed matter and quantum gas systems Chair: Renzo Ricca 13:30-14:10 Fleischer, JW (Princeton University) Superfluid and Berezinskii-Kosterlitz-Thouless dynamics of light Sem 1 There are many advantages to interpreting coherent light as a superfluid. From an optics perspective, fluid language gives new insight into old problems and leads to new physics, e.g. instabilities. From a physics perspective, the ability to control input conditions and directly image the output means that optical experiments enable the observation of features that are difficult, if not impossible, to see in other fields. This is particularly true of coherence dynamics, as phase relationships are relatively easy to uncover through interference. Here, we review our recent results on vortex dynamics and optical thermodynamics, with an emphasis on condensation phenomena and the BKT transition. In the former process, the approach to thermal equilibrium drives the largest-scale mode of the system to become macroscopically occupied. In the latter process, vortex generation becomes more favorable than entropy production, and attempts at long-range order are destroyed. These two processes compete yet can co-exist, with many aspects of their many-body physics still outstanding. Optical experiments are beginning to map the dynamical phase space, through direct measurement of energy and momentum density, vortex number, and coherence properties. While still in their early stages, the results demonstrate condensed matter physics using only light and reinforce the use of photonic systems as an experimental testbed for fluid and statistical physics. 14:10-14:30 Berloff, N (University of Cambridge) Vortices in non-equilibrium condensates Sem 1 14:30-14:50 Battye, R (University of Manchester) Vorton solutions in 2D and 3D Sem 1 We will present strong evidence for the existence of ring solutions in 2D and 3D known as vortons. In 2D we have are able to find analytic solutions for a twisted domain wall solutions which we use with a semi-analytic model to asset the existence of vortons and this is tested against numerical field theory solutions. We go on to the more difficult problem of vortons in 3D. We present solutions both in the cases of global and local U(1)xU(1) symmetries. 14:50-15:10 Yahalom, A (Ariel University Center of Samaria) Log-Analytic Uncertainty Relation Sem 1 I address the question of what characteristics of the quantum wave function phase can be measured. In particular I am interested in those phase aspects that can be deduced from the amplitude of the wave function. This will be shown to be connected to the topological characteristics of the Logarithm of the wave function in the complex time domain through a relation which is the temporal equivalent to the Kramers - Kronig formulae in the frequency domain. In particular for a wave packet which is reflected from an infinite potential barrier certain characteristics of the phase can be deduced provided that the momentum of the particle time the distance from the barrier is smaller than Planck constant over two and cannot be deduced otherwise. That is the phase characteristics cannot be deduced if the momentum of the particle time the distance from the barrier is bigger than Planck's constant over two which creates an experimental uncertainty in the phase. 15:30-15:50 Volkov, MS (University of Tours) Vortons and their stability Sem 1 We present classical field theory solutions describing stationary vortex loops carrying persistent currents and balanced against contraction by the centrifugal force. Objects of this type, sometimes called vortons, may exist in various physical contexts, ranging from domains of condensed matter physics described by the non-linear Schroedinger equation and/or Ginzburg-Landau equations, till models of high energy physics such as Witten's theory of superconducting cosmic strings. We also analyze the vorton stability, both at the linear and non-linear levels. 15:50-16:10 Pickett, G (Lancaster University) Investigating Pure Quantum Turbulence in Superfluid 3He and a Means of Directly Inferring the Turbulent Energy Content Sem 1 The lack of a general solution to the governing Navier-Stokes equations means that there is no fundamental theory of turbulence. Simpler pure quantum turbulence, the tangle of identical singly-quantized vortices in superfluids at T~0 may provide a deeper understanding of turbulence in general. In the present context this is especially relevant to the measurement of the turbulent energy. While the well-known Kolmogorov theory predicts the energy distribution of turbulence and how it decays, in normal systems the turbulent energy is generally only a small perturbation on the total thermal energy of the supporting medium. In quantum turbulence, however, the turbulent energy is accessible. A stationary condensate is necessarily in its ground state with zero enthalpy. Thus any added quantum turbulence accounts for the entire free energy of the superfluid and there are no other contributions. In superfluid 3He we can generated and detect quantum turbulence and how it evolves with time, and using bolometric methods, we can measure the energy released as previously-generated turbulence decays, which seems to be unique to these systems. 16:10-16:50 Couder, Y (Laboratoire Matière et Systèmes Complexes) Wave/Particle Duality via Silicon Droplets? Sem 1 16:50-17:10 Monastyrsky, M (ITEP (Institute for Theoretical and Experimental Physics)) Topology of Quantum Liquids Sem 1 15:10-15:30 Afternoon Tea
 Wednesday 5 December Session: Knots in mathematics: Knot energies Chair: Jason Cantarella 09:00-09:40 von der Mosel, H (RWTH Aachen University) Analytic and topological aspects of Menger curvatures for curves and submanifolds Sem 1 We discuss various types of geometric curvature energies based on the concept of Menger curvature. These energies exhibit self-avoidance and regularizing effects on curves and submanifolds, and they control their topology. 09:40-10:00 Mastin, M (University of Georgia) Symmetric Criticality for Ropelength Sem 1 The ropelength of a link embedded in $R^3$ is the ratio of the curve's length to its thickness. Jason Cantarella, Joe Fu, Rob Kusner, and John Sullivan have developed a theory of first order criticality for ropelength. We will discuss an extension of this work for the case of link conformations with rigid rotational symmetry. As an application we will prove that there is an infinite class of knots for which there are geometrically distinct ropelength critical conformations. This work is joint with Jason Cantarella, Jennifer Ellis, and Joe Fu. 10:00-10:20 Reiter, P (University of Duisburg-Essen) Regularity theory for knot energies Sem 1 In the past two decades, the introduction of several knot-based geometric functionals has greatly contributed to the field of geometric curvature energies. The general aim is investigating geometric properties of a given knotted curve in order to gain information on its knot type. More precisely, the original idea was to search a "nicely shaped" representative in a given knot class having strands being widely apart. This led to modeling self-avoiding functionals, so-called knot energies, that blow up on embedded curves converging to a curve with a self-intersection. Due to the singularities which guarantee the self-repulsion property all these functionals lead to interesting analytical problems which in many cases almost naturally involve fractional Sobolev spaces. In this talk we consider stationary points of knot energies. To this end we compute the Euler-Lagrange equation and derive higher regularity via a bootstrapping process. 10:20-10:40 Kusner, R (U Mass Amherst) Critical Links and Unlinks Sem 1 The configuration spaces Hopf_k or McCool_k of k-component Hopf-links or unlinks can be understood using extensions of Hatcher's proof of the Smale Conjecture. We will describe low-energy critical configurations for Möbius energy or Ropelength on Hopf_k or McCool_k, and sketch a picture of the bottom of the Morse-Smale complex for each. [This represents part of an ongoing project with Ryan Budney and John M. Sullivan] 11:00-11:40 O'Hara, J (Tokyo Metropolitan University) Renormalized potential energies and their asymptotics Sem 1 Energy of a knot was originally defined as the integration of the renormalized potential of a certain kind. Here, the renormalization can be done as follows: Suppose we are interested in a singular integral $\int_\Omega\omega$, which blows up on a subset $X\subset\Omega$. Remove an $\delta$-tubular neighbourhood of $X$ from $\Omega$, consider the integral over the complement, expand it in a Laurent series of $\delta$, and take the constant term. This idea gave rise to a M\"obius invariant surface energy in the sense of Auckly and Sadun, and recently, to generalization of Riesz potential of compact domains. If we integrate this generalized Riesz potential over the domain, we may need another renormalization around the boundary, according to the order of the generalized Riesz potential. In this talk I will give "baby cases" of the application of the above story to the study of knots or surfaces. 11:40-12:00 Buniy, RV (Chapman University) Higher order topological invariants from the Chern-Simons action Sem 1 12:00-12:20 Levina, G; Montgomery, M (Russian Academy of Sciences; NOAA/ Hurricane Research Division, Florida) Helical Organization of Tropical Cyclones Sem 1 Recently we found (Levina and Montgomery, 2010) that a tropical cyclone (TC) formation is accompanied by generation of essential nonzero and persistently increasing integral helicity. In this contribution we consider a helical flow organization on small and large space scales in a forming TC and offer a quantitative analysis for early stages in evolution of large-scale helical vortex based on diagnosis of a set of integral helical and energetic characteristics. Using the data from a near cloud-resolving numerical simulation, a key process of vertical vorticity generation from horizontal components and its amplification by special convective coherent structures – Vortical Hot Towers (VHTs) – is highlighted. The process is found to be a pathway for generation of a velocity field with linked vortex lines of horizontal and vertical vorticity on local and system scales. Based on these results, a new perspective on the role of VHTs in the amplification of the system-scale circulation is emphasized. They are THE CONNECTERS of the primary tangential and secondary overturning circulation on the system scales and are elemental building blocks for the nonzero system-scale helicity of the developing vortex throughout the TC evolution from genesis to the mature hurricane state. Calculation and analyses of helical and energetic characteristics together with hydro- and thermodynamic flow fields allow the diagnosis of tropical cyclogenesis as an event when the primary and secondary circulations become linked on system scales. We discuss also how these ideas may be combined with a recent paradigm of ‘Marsupial Pouch’ that allows predicting and tracking the location of tropical cyclogenesis in an easterly wave by means of global operational weather models. 12:20-12:35 Proment, D (University of East Anglia) Vortex knots in a Bose-Einstein condensate Sem 1 I will present a method for numerically building a quantum vortex knot state in the single scalar field wave function of a Bose-Einstein condensate. I will show how the two topologically simplest vortex knots wrapped over a torus evolve and may preserve their shapes by reporting results of the integration in time of the governing Gross-Pitaevskii equation. In particular, I will focus on how the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: in a first approximation it is linear and, for smaller ratio, the knot travels faster. Finally, I will display mechanisms of vortex breaking by reconnections which produce simpler vortex rings whose number depends on initial knot topology. 13:30-13:50 Salman, H (University of East Anglia) Solitons and Breathers on Quantized Superfluid Vortices Sem 1 It is well known that quantized superfluid vortices can support excitations in the form of helical Kelvin waves. These Kelvin waves play an important role in the dynamics of these vortices and their interactions are believed to be the key mechanism for transferring energy in the ultra low temperature regime of superfluid turbulence in $^4$He. Kelvin waves can be ascribed to low amplitude excitations on vortex filaments. In this talk I will show that larger amplitude excitations of the vortices can be attributed to solitons propagating along the vortex filament. I will review the different class of soliton solutions that can arise as determined analytically from a simplified vortex model based on the localized induction approximation. I will show, through numerical simulations, that these solutions persist even in more realistic models based on a vortex filament model and the Gross-Pitaevskii equation. As a generalisation of these soliton solutions, I also consider the breathe r solutions on a vortex filament and illustrate how, under certain conditions, large amplitude excitations that are localized in space and time can emerge from lower amplitude Kelvin wave like excitations. The results presented are quite generic and are believed to be relevant to a wide class of systems ranging from classical to superfluid vortices. I will also interpret our results on these nonlinear vortex excitations in the context of the cross-over regime of scales in superfluid turbulence. 13:50-14:10 Haley, R (Lancaster University) Branes, strings and boojums; topological defects in helium-3 and the cosmos Sem 1 The order parameter of the superfluid helium-3 condensate exhibits broken symmetries that show analogs with those broken in the various transitions undergone by the Universe after the Big Bang. Fortunately for us, the helium-3 order parameter is also sufficiently complex that the superfluid may exist in several phases, the two most stable being the A and B phases. At Lancaster we have developed techniques to investigate the properties of the interface between the A and B phases in the pure condensate limit, far below the superfluid transition temperature. The order parameter transforms continuously across the A–B boundary, making this interface the most coherent two-dimensional structure to which we have experimental access. It has been argued that this ordered 2-d surface in a 3-d bulk matrix, separating the two phases, can provide a good analog of a cosmological brane separating two distinct quantum vacuum states. In superfluid helium-3 the creation of such 2-branes mu st lead to the formation of point and line defects in the texture of the 3-d bulk, simply as a result of the constraints imposed by the interplay of the order parameter symmetries and the geometry of the container. Furthermore, our experiments have shown that removing the 2-branes from the bulk, in a process analogous to brane annihilation, creates new line defects in large quantities. Such observations may provide insight into the formation of topological defects such as cosmic strings arising from brane interactions in the early Universe. Up to now our experimental techniques have only allowed us to infer the properties of the interface and defects by measuring how they impede the transport of quasiparticle excitations in the superfluid, which is essentially a remote measurement. Our new experiments allow us to directly probe the interface region. 14:10-14:30 Jäykkä, J (Nordita) Dynamics of Hopfions Sem 1 Several materials, such as ferromagnets, spinor Bose-Einstein condensates and some topological insulators, are now believed to support knotted structures. One of the most successful base-models having stable knots is the Faddeev-Skyrme model and it is expected to be contained in some of these experimentally relevant models. The taxonomy of knotted topological solitons (Hopfions) of this model is known. In this talk, we describe the basic properties of static Hopfions, known for quite a long time before discussing some aspects of the dynamics of Hopfions, how the static properties survive in the dynamical situation, and show that they indeed behave like particles: during scattering the Hopf charge is conserved and bound states are formed when the dynamics allows it. 10:40-11:00 Morning Coffee 12:35-13:30 Lunch at Wolfson Court 14:30-17:00 Free Afternoon 19:30-22:00 Conference Dinner at Trinity Hall
 Thursday 6 December Session: Knots in mathematics: Tight Knots, etc. Chair: Rob Kusner 09:00-09:20 Parsley, J (Wake Forest University) Cohomology reveals when helicity is a diffeomorphism invariant Sem 1 We consider the helicity of a vector field, which calculates the average linking number of the field’s flowlines. Helicity is invariant under certain diffeomorphisms of its domain – we seek to understand which ones. Extending to differential (k+1)-forms on domains R^{2k+1}, we express helicity as a cohomology class. This topological approach allows us to find a general formula for how much helicity changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicity-preserving diffeomorphisms on a given domain, finding new ones on the two-holed solid torus and proving that there are no new ones on the standard solid torus. This approach also leads us to define submanifold helicities: differential (k+1)-forms on n-dimensional subdomains of R^m. 09:20-09:40 Shonkwiler, C (University of Georgia) The geometry of random polygons Sem 1 What is the expected shape of a ring polymer in solution? This is a natural question in statistical physics which suggests an equally interesting mathematical question: what are the statistics of the geometric invariants of random, fixed-length n-gons in space? Of course, this requires first answering a more basic question: what is the natural metric (and corresponding probability measure) on the compact manifold of fixed-length n-gons in space modulo translation? In this talk I will describe a natural metric on this space which is pushed forward from the standard metric on the Stiefel manifold of 2-frames in complex n-space via the coordinatewise Hopf map introduced by Hausmann and Knutson. With respect to the corresponding probability measure it is then possible to prove very precise statements about the statistical geometry of random polygons. For example, I will show that the expected radius of gyration of an n-gon sampled according to this measure is exactly 1/(2n). I will also demonstrate a simple, linear-time algorithm for directly sampling polygons from this measure. This is joint work with Jason Cantarella (University of Georgia, USA) and Tetsuo Deguchi (Ochanomizu University, Japan). 09:40-10:00 Cantarella, J (University of Georgia) The Expected Total Curvature of Random Polygons Sem 1 We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution. We then consider the symmetric measure on closed polygons of fixed total length constructed by Cantarella, Deguchi, and Shonkwiler. For this measure, the expected total curvature of a closed n-gon is asymptotic to n pi/2 + pi/4 by our first result. With a more careful analysis, we are able to prove that the exact expected value of total curvature is n pi/2 + (2n/2n-3) pi/4. As a consequence, we show that at least 1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in 3-space are unknotted. 10:00-10:20 Ricca, RL (Università degli Studi di Milano - Bicocca) On the energy spectrum of magnetic knots and links Sem 1 The groundstate energy spectrum of the first 250 zero-framed prime knots and links is studied by using an exact analytical expression derived by the constrained relaxation of standard magnetic flux tubes in ideal magneto-hydrodynamics (Maggioni & Ricca, 2009) and data obtained by the RIDGERUNNER tightening algorithm (Ashton et al., 2011). The magnetic energy is normalized with respect to the reference energy of the tight torus and is plotted against increasing values of ropelength. A remarkable generic behavior characterizes the spectrum of both knots and links. A comparative study of the bending energy reveals that curvature information provides a rather good indicator of magnetic energy levels. (2009) Maggioni F and Ricca RL. On the groundstate energy of tight knots. Proc. R. Soc. A 465, 2761–2783. (2011) Ashton T, Cantarella J, Piatek M and Rawdon E. Knot tightening by constrained gradient descent. Experim. Math. 20, 57-90. 10:20-10:40 Sullivan, JM (Technische Universität Berlin) Criticality theory for ropelength and related problems Sem 1 We consider certain new results related to the criticality theory for ropelength developed with Cantarella, Fu and Kusner. 11:10-11:30 Ernst, C (WKU) Topological and Geometric Properties of tightly confined random polygons Sem 1 Consider equilateral random polygons in a confinement sphere of radius R. In this talk we describe how geometric properties of the random polygons such a curvature or torsion change when the radius of confinement decreases. We also describe how the knot spectrum changes as R decreases. 11:30-11:45 Starostin, E (University College London) Equilibrium configurations of elastic torus knots (n,2) Sem 1 We study equilibria of braided structures made of two elastic rods with their centrelines remaining at constant distance from each other. The model is geometrically exact for large deformations. Each of the rods is modelled as thin, uniform, homogeneous, isotropic, inextensible, unshearable, intrinsically twisted, and to have circular cross-section. The governing equations are obtained by applying Hamilton's principle to the action which is a sum of the elastic strain energies and the constraints related to the inextensibility of the rods. Hamilton's principle is equivalent to the second-order variational problem for the action expressed in reduced strain-like variables. The Euler-Lagrange equations are derived partly in Euler-Poincare form and are a set of ODEs suitable for numerical solution. We model torus knots (n,2) as closed configurations of the 2-strand braid. We compute numerical solutions of this boundary value problem using path following. Closed 2-braids buckle under increasing twist. We present a bifurcation diagram in the twist-force plane for torus knots (n,2). Each knot has a V-shaped non-buckled branch with its vertex on the twist axis. There is a series of bifurcation points of buckling modes on both sides of each of the V-branches. The 1st mode bifurcation points for n and n±4 are connected by transition curves that go through (unphysical) self-crossing of the braid. Thus, all the knots turn out to be divided into two classes: one of them may be produced from the right-handed trefoil and the other from the left-handed. Higher-mode post-buckled configurations lead to cable knots. It is instructive to see how close our elastic knots can be tightened to the ideal shape. For the trefoil knot the tightest shape we could get has a ropelength of 32.85560666, which is remarkably close to the best current estimate. Careful examination reveals that the solution is free from self-intersections though the contact set remains a distorted circle. 11:45-12:05 Blatt, S (University of Warwick) The gradient flow of O'Hara's knot energies Sem 1 All of us know how hard it can be to decide whether the cable spaghetti lying in front of us is really knotted or whether the knot vanishes into thin air after pushing and pulling at the right strings. In this talk we approach this problem using gradient flows of a family of energies introduced by O'Hara in 1991-1994.We will see that this allows us to transform any closed curve into a special set of representatives - the stationary points of these energies - without changing the type of knot. We prove longtime existence and smooth convergence to stationary points for these evolution equations. 12:05-12:25 Tanda, S (Hokkaido University) Topological Crystals and the quantum effects Sem 1 We report the discovery of Mobius, Ring, Figure-8, Hopf-link Crystals in NbSe3, conventionally grown as ribbons and whiskers. We also reveal their formation mechanisms of which two crucial components are the spherical selenium (Se) droplet, which a NbSe3 ber wraps around due to surface tension, and the monoclinic (P2(1)/m) crystal symmetry inherent in NbSe3, which induces a twist in the strip when bent. Our crystals provide a non-fictious topological Mobius world governed by a non-trivial real-space topology. We classified these topological crystals as an intermediary between condensed matter physics and mathematics. Moreover, we observed Aharonov-Bohm effect of charge-density wave and Frolich type superconductor as electronic properties using topological crystals. 10:40-11:10 Morning Coffee 12:25-13:30 Lunch at Wolfson Court Session: Quantized flux tubes and vortices Chair: Robert Kerr 14:00-14:20 Bajer, K (Uniwersytet Warszawski) Quantum vortex reconnections Sem 1 14:20-14:40 Irvine, W (University of Chicago) Knots in light and fluids Sem 1 To tie a shoelace into a knot is a relatively simple affair. Tying a knot in a field is a different story, because the whole of space must be filled in a way that matches the knot being tied at the core. The possibility of such localized knottedness in a space-filling field has fascinated physicists and mathematicians ever since Kelvin’s 'vortex atom' hypothesis, in which the atoms of the periodic table were hypothesized to correspond to closed vortex loops of different knot types. An intriguing physical manifestation of the interplay between knots and fields is the possibility of having knotted dynamical excitations. I will discuss some remarkably intricate and stable topological structures that can exist in light fields whose evolution is governed entirely by the geometric structure of the field. A special solution based on a structure known as a Robinson Congruence that was re-discovered in different contexts will serve as a basis for the discussion. I will th en turn to hydrodynamics and discuss topologically non-trivial vortex configurations in fluids.My lab's website can be found at http://irvinelab.uchicago.edu 14:40-14:55 Proukakis, N (Newcastle University) Vortex Dynamics and Turbulence in Confined Quantum Gases Sem 1 Quantised vortices are known to arise in ultra-low temperature quantum gases as a result of targeted vortex generation (e.g. via phase imprinting or a 'quantum stirrer') or intrinsic system fluctuations. Such vortices interact dynamically, reconnect and can form regular ('lattices') or irregular (turbulent) structures, depending on the system conditions. Focusing initially on the issue of tangled vorticity, we show that the velocity statistics provides a unique identifier of 'quantum' vs. 'ordinary' turbulence, in agreement with related studies in helium. As quantum gas experiments typically feature harmonic confinement, one does not have access to the broad lengthscales relevant for helium, with the total number of vortices typically constrained from a few to a few hundred. In a first attempt to probe 'turbulence' in such systems, we go beyond the usual procedure of looking at the energy spectrum to discuss methods to quantify and ana lyze the amount of clustering of vortices using information extracted from their position and winding, focusing here on the two-dimensional regime. As realistic cold atom experiments are conducted at non-zero temperatures, where the condensate co-exists with a thermal cloud, we also study how temperature modifies the motion of vortices in such systems. This work has been generously funded by EPSRC. 14:55-15:10 Suramlishvilli, N (Newcastle University) Interpretation of quasiparticle scattering measurements in $^3$He-B: a three dimensional numerical analysis Sem 1 Present research is concerned with numerical modelling of Andreev scattering technique used for detection of quantized vortices in $^3$He-B. The results of numerical analysis of Andreev reflection by three-dimensional turbulent structures are reported. We analyse the Andreev scattering by a dense vortex tangle and calculate the spectral characteristics of the retroreflected beam of thermal excitations. The obtained results are in agreement with experimental observations. 15:30-15:45 Wacks, D (Newcastle University) Macroscopic bundles of vortex rings in superfluid helium Sem 1 It is well known that two co-axial vortex rings can leap-frog about each other. By direct numerical simulation, we show that in superfluid helium the effect can be generalised to a large number of vortex rings, which form a toroidal bundle. The bundle can be shown to be robust, travelling a significant distance compared to its diameter, whilst simultaneously becoming linked and eventually turbulent. We also discuss the effect of friction at non-zero temperatures, and show how in this case the presence of normal fluid rotation is necessary for the stability of the bundle. 15:45-16:05 Tkalec, U (Jozef Stefan Institute) Knots and links of disclination lines in chiral nematic colloids Sem 1 Nematic braids formed by disclination lines entangling colloidal particles in nematic liquid crystal are geometrically stabilized and restricted by topology. Experiments with nematic braids show rich variety of knotted and linked disclinations loops that can be manipulated and rewired by laser light. We describe a simple rewiring formalism and demonstrate how the self-linking number of nematic ribbons enables a classification of entangled structures. Controlled formation of arbitrary microscopic links and knots in nematic colloids provides a new route to the fabrication of soft matter with special topological features. References: [1] M. Ravnik, M. Škarabot, S. Žumer, U. Tkalec, I. Poberaj, D. Babič, N. Osterman, I. Muševič, Phys. Rev. Lett. 99, 247801 (2007). [2] U. Tkalec, M. Ravnik, S. Žumer, I. Muševič, Phys. Rev. Lett. 103, 127801 (2009). [3] S. Čopar, S. Žumer, Phys. Rev. Lett. 106, 177801 (2011). [4] U. Tkalec, M. Ravnik, S. Čopar, S. Žumer, I. Muševič, Science 333, 62 (2011). [5] S. Čopar, T. Porenta, S. Žumer, Phys. Rev. E 84, 051702 (2011). [6] G. P. Alexander, B. G. Chen, E. A. Matsumoto, R. D. Kamien, Rev. Mod. Phys. 84, 497 (2012). 16:05-16:25 Golov, A (University of Manchester) Experiments with tangles of quantized vortex lines in superfluid 4He in the T=0 limit Sem 1 In our experiments, we can create dense ensembles of quantized vortex lines of various degrees of polarization and entanglement, and monitor either their free decay or steady state whilst forcing continuously. The superfluid is forced either by macroscopic bodies that generate large-scale vortex bundles or by microscopic particles (injected ions) that generate uncorrelated vortices. Steady net polarization can be introduced by conducting the experiment in a rotating container. The characterization of vortex tangles is done via measurements of the transport of injected ions through them. The following types of tangles will be reviewed: homogeneous random tangles (no large-scale polarization), homogeneous quasi-classical turbulence (tangles in which the dominant energy is concentrated in large-scale bundles-eddies), steadily polarized anisotropic tangles of either high or low polarization, beams of parallel vortex rings. There is no viscous dissipation in the T=0 limit; the dyn amics of individual vortex lines is conservative except for the Kelvin waves of extremely small wavelengths. The scenario and rate of the evolution of different vortex ensembles largely depend on the mutual polarization of vortioces that affects the frequency of their reconnections. Further plans to investigate the microscopic processes of the quantum cascade (Kelvin wave cascade) and visualization of individual vortex cores will be outlined. 16:25-16:45 Bohr, J (DTU) Torus Knots and Links, Twist Neutrality and Biological Applications Sem 1 We present mathematical restrictions for torus knots and links, and for bent helices. The concept of twist neutrality is developed for bent and coiled structures and biological applications are reviewed [1,2]. [1] K. Olsen and J. Bohr, Geometry of the toroidal N-helix: optimal-packing and zero-twist. New Journal of Physics 14, 023063 (2012). [2] J. Bohr and K. Olsen, Twist neutrality and the diameter of the nucleosome core particle. Phys. Rev. Lett. 108, 098101 (2012). 15:10-15:30 Afternoon Tea
 Friday 7 December Session: Linking in magnetic fields and other physical systems Chair: Levon Pogosian 09:00-09:40 Vachaspati, T (ASU) Magnetic Fields in the Early Universe -- Chiral Effects and Topology Sem 1 I will briefly review cosmic magnetic fields and discuss some ideas to generate them. Special emphasis will be given to the possible generation of helical magnetic fields, and the possible role of chirality in the universe. As a by-product, the discussion will hint at processes that might lead to the production of magnetic monopoles. 09:40-10:20 Akhmet'ev, P (IZMIRAN) Asymptotic higher ergodic invariants of magnetic lines Sem 1 V.I.Arnol'd in 1984 formulated the following problem: "To transform asymptotic ergodic definition of Hopf invariant of a divergence-free vector field to Novikov's theory, which generalizes Withehead product in homotopy groups"'. We shall call divergence-free fields by magnetic fields. Asymptotic invariants of magnetic fields, in particular, the theorem by V.I.Arnol'd about asymptotic Gaussian linking number, is a bridge, which relates differential equitations and topology. We consider 3D case, the most important for applications. Asymptotic invariants are derived from a finite-type invariant of links, which has to be satisfied corresponding limit relations. Ergodicity of such an invariant means that this invariant is well-defined as the mean value of an integrable function, which is defined on the finite-type configuration space $K$, associated with magnetic lines. At the previous step of the construction we introduce a simplest infinite family of invariants: asymptotic linking coefficients. The definition of the invariants is simple: the helicity density is a well-defined function on the space $K$, the coefficients are well-defined as the corresponding integral momentum of this function. Using this general construction, a higher asymptotic ergodic invariant is well-defined. Assuming the the magnetic field is represented by a $\delta$-support with contains 3 closed magnetic lines equipped with unite magnetic flows, this higher invariant is equal to the corresponding Vassiliev's invariant of classical links of the order 7, and this invariant is not a function of the pairwise linking numbers of components. When the length of generic magnetic lines tends to $\infty$, the asymptotic of the invariant is equal to 12, this is less then twice order $14$ of the invariant. Preliminary results arXiv:1105.5876 was presented at the Conference "`Entanglement and Linking"' (Pisa) 18-19 May (2011). 10:40-11:20 Dunajski, M (University of Cambridge) Topological Solitons from Geometry Sem 1 Solitons are localised non-singular lumps of energy which describe particles non perturbatively. Finding the solitons usually involves solving nonlinear differential equations, but I shall show that in some cases the solitons emerge directly from the underlying space-time geometry: certain abelian vortices arise from surfaces of constant mean curvature in Minkowski space, and skyrmions can be constructed from the holonomy of gravitational instantons. 10:20-10:40 Morning Coffee Session: Topological Solitons Chair: Konrad Bajer 11:20-11:40 Nitta, M (Keio University) Creating Vortons and Knot Solitons via Domain Wall Pair Annihilation in BEC and Field Theory Sem 1 We show that when a vortex-string is stretched between a pair of a domain wall and an anti-domain wall in two component Bose-Einstein condensates, there remains a vorton after the pair annihilation. We also show that the same configuration in the mass deformed Faddeev-Skyrme model results in a knot soliton (Hopfion) after the pair annihilation. 11:40-12:00 Tchrakian, T (DIAS) Abelian and non-Abelian Hopfions in all odd dimensions Sem 1 Hopfions are field configurations of scalar matter systems characterised prominently by the fact that they describe knots in configuration space. Like the 'usual solitons', e.g. Skyrmions, monopoles, vortices and instantons, Hopfions are static and finite energy solutions that are stabilised by a topological charge, which supplies the energy lower bound. In contrast to the 'usual solitons' however, the topological charge of Hopfions is not the volume integral of a total divergence. While the topological charge densities of the 'usual solitons', namely the Chern-Pontryagin (CP) densities or their descendants, are total divergence, the corresponding quantities for Hopfions are the Chern-Simons (CS) densities which are not total divergence. Subject to the appropriate symmetries however, these CS densities do reduce to total divergence and become candidates for topological charges. Thus, Hopfion field are necessarily subject to the appropriate symmetry to decsribe knots, excluding spherically symmetry, in contrast to the 'usual solitons'. The construction of these CS densities is enabled by employing complex nonlinear sigma models, which feature composite connections. The CS densities are defined in terms of these connections and their curvatures. (In some dimensions the complex sigma model can be equivalent to a real sigma model, e.g. in D=3 Skyrme-Fadde'ev O(3) model and the corresponding CP^1 model.) It is natural to propose Hopfion fields in all odd space dimensions where a CS density can be defined. This covers both Abelian and non-Abelian theories, namely empolying projective-complex and Grassmannian models, respectively. It is in this sense that we have used the terminology of Abelian and non-Abelian Hopfions. Explicit field configurations displaying the appropriate symmetries and specific asymptotic behaviours in several (higher) dimensions are proposed, and it is verified that for these configurations the CS densities do indeed become total divergence. 12:00-12:20 Babaev, E (UMass Amherst and KTH Stockholm) Skyrmions and Hopfions in exotic superconductors Sem 1 12:30-13:30 Lunch at Wolfson Court Session: Topological Solitons continued Chair: Tom Kephart 13:30-13:50 Harland, D (Loughborough University) Modelling Hopf solitons with elastic rods Sem 1 I will review recent progress in modelling knotted solitons in the Skyrme-Faddeev model using elastic rods. The effective elastic rod model is simple to use, and can in some cases be solved analytically. It has enabled the discovery of new solitonic states which had eluded direct numerical simulations of the field theory. This suggests more generally that elasticity theory could be a useful tool in the study of solitons. 13:50-14:10 Speight, M (University of Leeds) Fermionic quantization of knot solitons Sem 1 Knot solitons arise as global energy minimizers in field theories such as the Faddeev-Skyrme model. Such field theories, when quantized, are inherently bosonic because the fundamental fields represent scalar bosons. Nonetheless, the solitons they support can be given fermionic exchange statistics, provided the classical field configuration space has the right algebraic topology. In this talk I will review a computation of the fundamental group of the Faddeev-Skyrme configuration space, and show how this allows a consistent fermionic quantization of knot solitons. This is based on (separate) collaborations with Dave Auckly and Steffen Krusch. 14:30-14:50 Niemi, A (CNRS/Uppsala University) Folding and collapse in string-like structures Sem 1 We argue that the the physics of folding and collapse of string-like structures can be described in terms of topological solitons. For this we use extrinsic geometry of filamental curves in combination of general geometrical arguments, to derive a universal form of energy function, which we propose is essentially unique. We then show that the ensuing equations of motion support topological solitons that are closely related to the solitons in the discrete nonlinear Schrodinger equation. We then argue that with proper parameters, a soliton supporting filament can describe proteins, which are mathematically one dimensional piecewise linear polygonal chains. As an example we show a movie of a simulation, how to model the folding of a medium length protein. The precision we reach is around 40 pico-meters root-mean-square distance form the experimentally constructed structure. Our result proposes that there are at least some 10^20 topological solitons in each human body. 14:50-15:10 Krusch, S (University of Kent) Fermions coupled to Hopf Solitons Sem 1 Solitons in the Skyrme and the Faddeev-Skyrme model share many similarities. While no topologically non-trivial exact solutions are known in flat space there is a minimal energy charge one soliton on the 3-sphere of sufficiently small radius in both models. The charge one Skyrmion is given by the identity map, whereas the charge one Hopf soliton is given by the Hopf map. Also, the solitons in both models can be semi-classically quantized as fermions, by defining the wave function of the covering space of configuration space and imposing Finkelstein-Rubinstein constraints. When fermions are chirally coupled to Skyrmions the resulting Dirac equation can be solved explicitly on the 3-sphere in the background of a charge one Skyrmion. In this talk, I describe how to couple fermions to Hopf solitons. 15:10-15:25 Garaud, J (UMass Amherst and KTH Stockholm) Topological solitons in multi-component superconductors : from baby-Skyrmions to vortex loops Sem 1 The crucial importance of topological excitations in the physics of superconductivity made Ginzburg--Landau vortices one of the most studied example of topological defects. Multi-band/multi-component superconductors extends usual Ginzburg--Landau theory by considering more than one scalar field (several superconducting order parameters). Family materials, where superconductivity is multi-band/multi-component, has recently been growing. Because of additional fields and new broken symmetries, the zoo of topological defects is much richer in multi-component systems (e.g. Line-like vortices, fractional vortices, baby skyrmions...). For entropic reasons, thermal fluctuations will induce much more complicated three dimensional solitons, in particular vortex loops. I will discuss various aspects of topological excitations in multi-component/multi-band superconductors. 14:10-14:30 Afternoon Tea 15:25-15:40 Closing Remarks