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Line defects are one-dimensional phase singularities (forming knots and links) that arise in a variety of physical systems. In these systems, isophase surfaces (Seifert surfaces) have the phase defects as boundary, and these Seifert surfaces define a framing of the normal bundle of each link component. We define the individual helicity for each component of a link singularity, and prove that each individual helicity is zero if and only if there exists a Seifert framing for the link. We extend these results to multi-armed defects. We prove that under anti-parallel reconnection of defect strands total twist is conserved.

In this paper we determine the effects of winding number on the dynamics of vortex torus knots and unknots in the context of classical, ideal fluid mechanics. We prove that the winding number — a topological invariant of torus knots — has a primary effect on vortex motion. This is done by applying the Moore-Saffman desingularization technique to the full Biot-Savart induction law, determining the influence of winding number on the 3 components of the induced velocity. Results have been obtained for 56 knots and unknots up to 51 crossings. In agreement with previous numerical results we prove that in general the propagation speed increases with the number of toroidal coils, but we notice that the number of poloidal coils may greatly modify the motion. Indeed we prove that for increasing aspect ratio and number of poloidal coils vortex motion can be even reversed, in agreement with previous numerical observations. These results demonstrate the importance of three-dimensional features in vortex dynamics and find useful applications to understand helicity and energy transfers across scales in vortical flows.

Topological fluid mechanics is a crucial area of classical mechanics, with remarkable theoretical significance and wide applications in practice. In this paper, we expect to present an introductory review of the direction, for the purpose of attracting more domestic researchers of China to enter into this important area. Our emphasis is placed on the topological essence of fluid helicity. We will give the relationship between helicity and mathematical knotted field theory (i.e., that between helicity and the mutual- and self-linking numbers, as well as the fluid knot polynomial topological invariants constructed in terms of helicity recently discovered by the authors), and give an introduction to the international research on energy-structural complexity relationship for a fluid vortex knot ensemble. Moreover, typical numerical methods are also demonstrated through examples of reconnections occurring in superfluid vortex knots/links. Expectedly, a combination of the theoretical framework and numerical techniques may contribute to the reader a comprehensive understanding of the main target, research methodology and potential technical difficulties in practice in this interdisciplinary field of cutting-edge research world-wide.

In this paper we show that twist, defined in terms of rotation of the phase associated with quantum vortices and other physical defects effectively deprived of internal structure, is a property that has observable effects in terms of induced axial flow. For this we consider quantum vortices governed by the Gross–Pitaevskii equation (GPE) and perform a number of test cases to investigate and compare the effects of twist in two different contexts: (i) when this is artificially superimposed on an initially untwisted vortex ring; (ii) when it is naturally produced on the ring by the simultaneous presence of a central straight vortex. In the first case large amplitude perturbations quickly develop, generated by the unnatural setting of the initial condition that is not an analytical solution of the GPE. In the second case much milder perturbations emerge, signature of a genuine physical process. This scenario is confirmed by other test cases performed at higher twist values. Since the second setting corresponds to essential linking, these results provide new evidence of the influence of topology on physics.

By numerically solving the three-dimensional Gross-Pitaevskii equation we analyze the cascade process associated with the evolution and decay of a pair of linked vortex rings. The system decays through a series of reconnections to produce finally three unlinked, unfolded, almost planar vortex loops. Total helicity, initially zero, remains unchanged throughout the process. The gradual transfer from writhe (due to initial linking) to twist helicity, followed by a continuous relaxation of twist across scales during the evolution is shown to be a generic mechanism that consistently takes place on each individual component.

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.

By considering steady magnetic fields in the shape of torus knots and unknots in ideal magnetohydrodynamics, we compute some fundamental geometric and physical properties to provide estimates for magnetic energy and helicity. By making use of an appropriate parametrization, we show that knots with dominant toroidal coils that are a good model for solar coronal loops have negligible total torsion contribution to magnetic helicity while writhing number provides a good proxy. Hence, by the algebraic definition of writhe based on crossing numbers, we show that the estimated values of writhe based on image analysis provide reliable information for the exact values of helicity. We also show that magnetic energy is linearly related to helicity, and the effect of the confinement of magnetic field can be expressed in terms of geometric information. These results can find useful application in solar and plasma physics, where braided structures are often present.

Here we show that under quantum reconnection, simulated by using the three-dimensional Gross-Pitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that the total length of the vortex system reaches a maximum at the reconnection time, while both writhe helicity and twist helicity remain separately unchanged throughout the process. Self-helicity is computed by two independent methods, and topological information is based on the extraction and analysis of geometric quantities such as writhe, total torsion, and intrinsic twist of the reconnecting vortex rings.

Reconnection is a fundamental event in many areas of science, from the interaction of vortices in classical and quantum fluids, and magnetic flux tubes in magnetohydrodynamics and plasma physics, to the recombination in polymer physics and DNA biology. By using fundamental results in topological fluid mechanics, the helicity of a flux tube can be calculated in terms of writhe and twist contributions. Here we show that the writhe is conserved under anti-parallel reconnection. Hence, for a pair of interacting flux tubes of equal flux, if the twist of the reconnected tube is the sum of the original twists of the interacting tubes, then helicity is conserved during reconnection. Thus, any deviation from helicity conservation is entirely due to the intrinsic twist inserted or deleted locally at the reconnection site. This result has important implications for helicity and energy considerations in various physical contexts., Comment: 21 pages, 4 figures

In this paper we derive and compare numerical sequences obtained by adapted polynomials such as HOMFLYPT, Jones and Alexander-Conway for the topological cascade of vortex torus knots and links that progressively untie by a single reconnection event at a time. Two cases are considered: the alternate sequence of knots and co-oriented links (with positive crossings) and the sequence of two-component links with oppositely oriented components (negative crossings). New recurrence equations are derived and sequences of numerical values are computed. In all cases the adapted HOMFLYPT polynomial proves to be the best quantifier for the topological cascade of torus knots and links.

By making simple, heuristic assumptions, a new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial expressed in terms of kinetic helicity. Then, for the sake of illustration, explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knot and for the Whitehead link via the figure-of-eight knot are considered. The resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. While this heuristic approach extends the use of helicity in terms of linking numbers to the much richer context of knot polynomials, it gives also rise to new interesting problems in mathematical physics and it offers new tools to perform real-time numerical diagnostics of complex flows.

In this paper we examine certain geometric and topological aspects of the dynamics and energetics of vortex torus knots and un-knots. The knots are given by small-amplitude torus knot solutions in the local induction approximation (LIA). Vortex evolution is then studied in the context of the Euler equations by direct numerical integration of the Biot-Savart law and the velocity, helicity and kinetic energy of different vortex knots and unknots are presented for comparison. Vortex complexity is parametrized by the winding number w given by the ratio of the number of meridian wraps to 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 the same size and circulation, whereas for w > 1, knots and poloidal coils have approximately the same speed and energy as the reference vortex ring. Kinetic helicity is dominated by writhe contributions and increases with knot complexity. All torus knots and unknots tested under Biot-Savart show much stronger permanence than under LIA.

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.

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.

The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Călugăreanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Călugăreanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Călugăreanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.

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 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 the context of the localized induction approximation (LIA) for the motion of a thin vortex filament in a perfect fluid, the present work deals with certain conserved quantities that emerge from the Betchov-Da Rios equations. Here, by showing that these invariants belong to a countable family of polynomial invariants for the related nonlinear Schrödinger equation (NLSE), it is demonstrated how to interpret them in terms of kinetic energy, pseudohelicity, and associated Lagrangian. It is also shown that under LIA both linear momentum and angular momentum are conserved quantities and the relation between these quantities and the whole family of polynomial invariants is discussed. © 1992 American Institute of Physics.

New measures of algebraic, geometric and topological complexity are introduced and tested to quantify morphological aspects 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 measurements 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.