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In this article we present a review of some of the author's most recent results in topological magnetohydrodynamics (MHD), with an eye to possible applications to astrophysical flows and solar coronal structures. First, we briefly review basic work on magnetic helicity and linking numbers, and fundamental relations with magnetic energy and average crossing numbers of magnetic systems in ideal conditions. In the case of magnetic knots, we focus on the relation between their groundstate energy and topology, discussing the energy spectrum of tight knots in terms of ropelength. We compare this spectrum with the one given by considering the bending energy of such idealized knots, showing that curvature information provides a rather good indicator of magnetic energy contents. For loose knots far from equilibrium we show that inflexional states determine the transition to braid form. New lower bounds for tight knots and braids are then established. We conclude with results on energy-complexity relations for systems in presence of dissipation.

In this paper we prove that under ideal conditions the helicity of fluid knots, such as vortex filaments or magnetic flux tubes, provides a fundamentally new topological means by which wemay associate a topological invariant, the Jones polynomial, that is much stronger than prior interpretations in terms of Gauss linking numbers. Our proof is based on an extension of the Kauffman bracket polynomial for unoriented knots. Explicit calculations of the Jones polynomial for the left- and right-handed trefoil knots and for the Whitehead link via the figure-of-eight knot are presented for illustration. This novel approach establishes a topological foundation of classical field theory in general, and of mathematical fluid dynamics in particular, by opening up new directions of work both in theory and applications

In this paper we present an application of a new technique, based on recent work done by Liu & Ricca (2012), to quantify structural complexity by means of topological methods. These rely on the derivation of the Jones polynomial from the helicity of ideal fluid flows. The techniques discussed here can be extended and applied to real fluid flows subject to continuous topological restructuring.

Structural complexity emerges from all systems that display morphological organization. Structural complexity for an intricate tangle of filaments is measured by the average crossing number of the tangle. Direct relationships between total length, energy, and structural complexity of a tangle are established. These results are based on elementary considerations that suggest a wide range of applications from magnetic energy estimates to neural and social networks and financial markets analysis.

In this paper we shall review some of the most recent developments and results on work on energy-complexity relations and, if time will allow it, we shall provide an analytical proof of eq. (3) below, a fundamental relation between energy and complexity established by numerical experiments.

In these lectures we present for the first time a mathematical reconstruction of what might have been Gauss' \textit{own} derivation of the linking number of 1833, and we provide also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction offered here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism and it is complemented by the independent analysis and discussion made by Maxwell in 1867. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important r\^ole in modern mathematical physics, we offer a direct proof of their equivalence, and we provide some examples of linking number computation based on oriented area information. The material presented in these lectures forms an integral part of a paper that will appear in the Journal of Knot Theory and Its Ramifications.

In this paper we provide a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction presented here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism. A brief discussion of a possibly independent derivation made by Maxwell in 1867 completes this reconstruction. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important role in modern mathematical physics, we offer a direct proof of their equivalence. Explicit examples of its interpretation in terms of oriented area are also provided.

In this paper we determine the velocity, the energy, and estimate writhe and twist helicity contributions of vortex filaments in the shape of torus knots and unknots (as toroidal and poloidal coils) in a perfect fluid. Calculations are performed by numerical integration of the Biot-Savart law. Vortex complexity is parametrized by the winding number w given by the ratio of the number of meridian wraps to that of longitudinal wraps. We find that for w<1 vortex knots and toroidal coils move faster and carry more energy than a reference vortex ring of same size and circulation, whereas for w>1 knots and poloidal coils have approximately same speed and energy of the reference vortex ring. Helicity is dominated by writhe contributions. Finally, we confirm the stabilizing effect of the Biot-Savart law for all knots and unknots tested, found to be structurally stable over a distance of several diameters. Our results also apply to quantized vortices in superfluid H4 e. © 2010 The American Physical Society.

It is well-known that a soap film spanning a looped wire can have the topology of a Möbius strip and that deformations of the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained unknown. Experimental studies presented here show that this process consists of a collapse of the film toward the boundary that produces a previously unrecognized finite-time twist singularity that changes the linking number of the film’s Plateau border and the centerline of the wire. We conjecture that it is a general feature of this type of transition that the singularity always occurs at the surface boundary. The change in linking number is shown to be a consequence of a viscous reconnection of the Plateau border at the moment of the singularity. High-speed imaging of the collapse dynamics of the film’s throat, similar to that of the central opening of a catenoid, reveals a crossover between two power laws. Far from the singularity, it is suggested that the collapse is controlled by dissipation within the fluid film surrounding the wire, whereas closer to the transition the power law has the classical form arising from a balance between air inertia and surface tension. Analytical and numerical studies of minimal surfaces and ruled surfaces are used to gain insight into the energetics underlying the transition and the twisted geometry in the neighborhood of the singularity. A number of challenging mathematical questions arising from these observations are posed.

With this paper we want to pay tribute to 150 years of work on topological fluid mechanics. For this, we review Helmholtz's (1858) original contribution on topological issues related to vortex motion. Some recent results on aspects of structural complexity analysis of fluid flows are presented and discussed, as well as new results on topological bounds on the energy of magnetic knots and links in ideal magnetohydrodynamics, and on helicity-crossing number relations in dissipative fluids. © 2009 Springer-Verlag Berlin Heidelberg.

From visual inspection of complex phenomena to modern visiometrics, the quest for relating aspects of structural and morphological complexity to hidden physical and biological laws has accompanied progress in science ever since its origin. By using concepts and methods borrowed from differential and integral geometry, geometric and algebraic topology, and information from dynamical system analysis, there is now an unprecedented chance to develop new powerful diagnostic tools to detect and analyze complexity from both observational and computational data, relating this complexity to fundamental properties of the system. In this paper we briefly review some of the most recent developments and results in the field. We give some examples, taken from studies on vortex entanglement, topological complexity of magnetic fields, DNA knots, by concluding with some comments on morphological analysis of structures present as far afield as in cosmology and brain research. © Springer-Verlag Italia, Milano 2009.

New results on the kinetic energy of ideal vortex filaments in the shape of torus knots and unknots are presented. These knots are given by smallamplitude torus knot solutions (Ricca, 1993) to the Localized Induction Approximation (LIA) law. The kinetic energy of different knot and unknot types is calculated and presented for comparison. These results provide new information on relationships between geometry, topology and dynamics of complex vortex systems and help to establish possible connections between

In this paper we present and discuss ideas and new results in three different research areas of topological fluid mechanics. First, we propose a conjectured experiment to produce and observe, for the first time, vortex knotting in real fluids. Next we provide a new ropelength bound for tight, magnetic knots in ideal magnetohydrodynamics. Finally, we present a novel interpretation of eigenvalue analysis of tensor fields in terms of integral geometry by using the information on form factors provided by structural complexity analysis

New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume- and flux-preserving flow, with the additional constraints that the tube cross section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm. Results for torus knots are compared with previous work, and the groundstate energy spectrum of the first prime knots — up to 10 crossings — is presented and analysed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration

A geometric method based on information from structural complexity is presented to calculate linear and angular momenta of a tangle of vortex filaments in Euler flows. For thin filaments under the so-called localized induction approximation the components of linear momentum admit interpretation in terms of projected area. By computing the signed areas of the projected graph diagrams associated with the vortex tangle, we show how to calculate the two momenta of the system by complexity analysis of tangle diagrams. This method represents a novel technique to extract dynamical information of complex systems from geometric and topological properties and provides a potentially useful tool to test the accuracy of numerical methods and investigate scale distribution of fluid dynamical properties of vortex flows.

In this paper, we determine two quantities, of geometric and topological character, that were left undetermined in two previous results obtained by Arnold ( Arnold 1974 In Proc. Summer School in Diff. Eqs. at Dilizhan, pp. 229 256.) and Moffatt ( Moffatt 1990 Nature 347, 367 - 369) on lower bounds for the magnetic energy of knots and links in ideal fluids. For dissipative systems, a lower bound on magnetic helicity in terms of the average crossing number and a new relationship between rates of change of these two quantities are also determined.

In this paper we introduce and analyze a set of equations to study geometric and energetic aspects associated with the kinematics of multiple folding and coiling of closed filaments for DNA modeling. By these equations we demonstrate that a high degree of coiling may be achieved at relatively low energy costs through appropriate writhe and twist distribution, and independently from the number of coils formed. For sufficiently high twist we show that coiling is actually favoured by elastic energy relaxation, when the deformation energy is due to curvature and the mean twist of the filament. We give estimates for the writhing process based purely on curvature information and the number of coils produced. We also determine the packing rate associated with filament compaction in the case of a hierarchical helical coiling. These results find useful applications in DNA biology, especially in modeling DNA wrapping in proteic regions, where there is a strong connection between high coiling, efficient compaction and energy localization.

Preliminary results on a new Stretch-Twist-Fold (STF) kinematic model for fast dynamo are presented. The evolution is prescribed by equations that govern the simultaneous stretching, writhing and coiling of a magnetic flux-tube by diffeomorphism of the initial circular configuration. Simple estimates based on minimized magnetic energy show that exponential growth of the magnetic field is indeed possible.

New kinematic equations are used to model DNA supercoiling. These equations govern the simultaneous production of multiple folding and coiling of a filament in space by diffeomorphism of a reference curve. Here we show how to use these equations to model regions of highly localized filament coiling, where mechanisms of proteic coding or viral spooling are important.

The kinematics of writhing and coiling of circular filaments is here analysed by new equations that govern the evolution of curves generated by epicycloids and hypocycloids. We show how efficiency of coil formation and compaction depend on writhing rates, relative bending, torsion and mean twist energy. We demonstrate that for coiling formation hypocycloid evolution achieves higher writhing rates, but in terms of deformation energy, the epicycloid evolution is much more effective. We also show how the occurrence and multiple appearance of inflexional configurations determine coil formation. Compaction and packing rate are also briefly examined. These results are fundamental and provide useful information for physical applications and for modelling natural phenomena, including relaxation of magnetic fields in the solar corona, magnetic dynamos in astrophysical flows, tertiary folding of macromolecules in chemical-physics, and DNA packing in cell biology. © 2006 The Royal Society.

In this paper, we prove that magnetic flux-tubes in inflexional configuration are in disequilibrium and evolve to an inflexion-free state. The magnetic field is defined in a tube of circular cross-section and is chosen so as to have toroidal and poloidal components contributing to the internal twist of the flux-tube. By using orthogonal curvilinear coordinates, we derive the equations for the Lorentz force associated with the magnetic flux-tube and use them to study the generic behaviour associated with passage through inflexional configuration. Inflexional state is attained when local curvature vanishes as the tube axis changes concavity. We check that the conditions for the equilibrium of the flux-tube in inflexional configuration are not satisfied and hence we prove disequilibrium. Inflexional disequilibrium makes magnetic flux-tubes evolve to an inflexion-free state. As a consequence, inflexional flux-tubes in free space evolve naturally to an inflexion-free configuration. This mechanism has important implications for the energetics of solar coronal loops and astrophysical flows. © 2005 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.

In this paper we introduce and discuss new concepts useful to analyse and characterize patterns of vortex lines in fluid flows. We define measures of tropicity to identify ‘tubeness’, ‘sheetness’ and ‘bulkiness’ of vortex lines and to measure the spreading of field lines about preferred directions. Algebraic, geometric and topological measures based on crossing number information are discussed and are put in relation to the kinetic helicity and the energy of the fluid system.

In this paper we review some results on geometric and topological vortex dynamics. After some background on flow maps, topological equivalence of frozen fields and conservation laws, we discuss geometric aspects of vortex filament motion (intrinsic equations, connections with integrable dynamics and extension to higher dimensional manifolds) and the topological interpretation of kinetic helicity in terms of linking numbers. We recall basic results on evolution of vortex knots and links and outline possible applications of algebraic, geometric and topological measures to evaluate structural complexity of vortex flows

New measures of algebraic, geometric and topological complexity are introduced and tested to quantify morphological as- pects of a generic tangle of filaments. The tangle is produced by standard numerical simulation of superfluid helium turbulence, which we use as a benchmark for numerical investigation of complex systems. We find that the measures used, based on crossing number information, are good indicators of generic behaviour and detect accurately a tangle’s complexity. Direct measure- ments of kinetic helicity are found to be in agreement with the other complexity-based measures, proving that helicity is also a good indicator of structural complexity. We find that complexity-based measure growth rates are consistently similar to one another. The growth rate of kinetic helicity is found to be twice that of energy.

In this paper we show how new techniques of topological fluid mechanics and physical knot theory can be applied to estimate magnetic energy levels in solar physics. In particular, we show that magnetic energy stored in complex configurations of plasma loops present on the Sun can be quantified by geometric and topological information. These studies find important applications in the energetics of solar and stellar physics and in laboratory plasmas.

For the first time since Lord Kelvin's original conjectures of 1875 we address and study the time evolution of vortex knots in the context of the Euler equations. The vortex knot is given by a thin vortex filament in the shape of a torus knot T(p,q) (p > 1, q > 1; p, q co-prime integers). The time evolution is studied numerically by using the Biot-Savart (BS) induction law and the localized induction approximation (LIA) equation. Results obtained using the two methods are compared to each other and to the analytic stability analysis of Ricca (1993, 1995). The most interesting finding is that thin vortex knots which are unstable under the LIA have a greatly extended lifetime when the BS law is used. These results provide useful information for modelling complex structures by using elementary vortex knots

In this paper we present new results concerning the evolution and stability of vortex knots in the context of the Euler equations. For the first time, since Lord Kelvin’s original conjecture of 1875, we have direct numerical evidence of stability of vortex filaments in the shape of torus knots. The results are based on the analytical solutions of Ricca [1] for thin vortex filaments and numerical integration of the Biot-Savart induction law. Moreover, a comparative study of vortex knot evolution under the so-called Localized Induction Approximation (LIA), which is a low-order approximation to the Biot-Savart law, confirms the stability results predicted by the LIA analysis. In particular, we show that thin vortex knots which are unstable under LIA have a greatly extended lifetime when the Biot-Savart law is used, but thick vortex knots have the same stability behaviour for both equations of motion

We have investigated numerically the motion and the stability of quantized vortex knots.

This paper presents new results concerning evolution and inflexional instability of twisted magnetic flux tubes in the solar corona. Inflexional configurations, attained when the curvature of the tube axis vanishes, are generally present in coronal magnetic structures and are invariably associated with the early stages of kink formation. New equations for the Lorentz force in orthogonal curvilinear coordinates are applied to study the behaviour of twisted flux tubes in presence of inflexion points. We find that inflexional flux tubes are in disequilibrium and evolve spontaneously to inflexion-free configurations, possibly in braid form. These results have important applications for solar coronal structures. First, they prove that the evolution and relaxation of twisted magnetic fields into braid form is a generic feature, confirming the observational evidence of highly twisted and braided structures present in the solar corona. Secondly, they demonstrate that inflexions can trigger kink instabilities, providing a fundamental mechanism for modeling outbreaks of energy into heat, emitted by flares, microflares and mass ejections.

In this paper we present for the first time a detailed account of the work of L.S. Da Rios and T. Levi-Civita on what is believed to be one of the first major contributions to three-dimensional vortex filament dynamics. Their work spanned a period of almost 30 years, from 1906 to 1933, and despite many publications remained almost unnoticed throughout this century. After a partial re-discovery (Ricca, 1991a), new material has now been found and is presented here with a full review of their work in relation to the present state of the art in non-linear mechanics and vortex dynamics. Their results include the conception of the localized induction approximation (LIA) for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of motion, the asymptotic potential theory applied to vortex tubes, the derivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations and the stability analysis of circular vortex filaments. In the light of modern developments in non-linear fluid mechanics, their work strikes for modernity and depth of results. Even more striking is the fact that this work remained obscure for almost a century. The results of Da Rios are particularly important in the study of integrable one-dimensional systems and vortex filament motion; Levi-Civita's work on asymptotic potential for slender tubes is at the core of the mathematical formulation of potential theory and capacity theory.

Geometric and topological aspects associated with integrability of vortex filament motion in the Localized Induction Approximation (LIA) context (which includes a family of local dynamical laws) are discussed. We show how to interpret integrability in relation to the Biot-Savart law and how soliton invariants can be interpreted in terms of global geometric functionals of knotted solutions. Under the basic (zeroth-order) LIA, we prove that vortex filaments in the shape of torus knots T p, q (p, q co-prime) with (q/p)>1 are stable, whereas those with (q/p)<1 are unstable.

A full investigation of the energy spectrum of a twisted flexible string under elastic relaxation is presented and discussed in detail for the first time. New polynomial expressions for critical energy states are derived and the whole spectrum of critical states (minima, maxima and inflexion points) of the elastic energy is found and discussed in relation to superhelicity and elastic characteristics of the string. The study is carried out in the context of the theory of linear elasticity and the thin rod approximation. The relaxation mechanism is studied by using conservation of linking difference (by the formula Delta Lk=Wr+ Delta Tw). We show how specific geometric quantities, such as pitch angle, writhe and twist contributions, as well as physical quantities, such as torsional and bending energy, depend on superhelicity (given by the specific linking difference) and elastic characteristics of the string (given by bending and torsional rigidity). These quantities, expressed per unit length, are examined and compared at each critical energy state. Starting from a supertwisted configuration, we show that the string relaxes (by twist reduction) through two different intermediate helical states (which correspond to different local minima), to reach the lowest minimum energy state in a supercoiled configuration. The case of a generic kink formation (and consequent passage through an inflexional configuration) is then examined and new expressions for the energy change in the vicinity of the inflexion point are derived.

In this paper we analyse in detail, and for the first time, the rôle of torsion in the dynamics of twisted vortex filaments. We demonstrate that torsion may influence considerably the motion of helical vortex filaments in an incompressible perfect fluid. The binormal component of the induced velocity, asymptotically responsible for the displacement of the vortex filament in the fluid, is closely analysed. The study is performed by applying the prescription of Moore & Saffman (1972) to helices of any pitch and a new asymptotic integral formula is derived. We give a rigorous proof that the Kelvin régime and its limit behaviour are obtained as a limit form of that integral asymptotic formula. The results are compared with new calculations based on the re-elaboration of Hardin's (1982) approach and with results obtained by Levy & Forsdyke (1928) and Widnall (1972) for helices of small pitch, here also re-elaborated for the purpose.