Same maths, different names · Audit 12
Knot two smoke rings through each other and an ideal fluid will never let them come apart. Thread two tubes of magnetic field through each other in a perfectly conducting plasma and the same is true. The number that records the threading is called helicity in both fields, and the same functional, or a construction built from it, appears again under other names in the topology of knots, in the supercoiling of DNA, in the quantum Hall effect and in the theory that turned knot invariants into a quantum field theory. This audit records that one helicity functional, a potential paired with its own flux, appears exactly in fluids, in plasmas and in abelian Chern–Simons theory, and that the other fields reach it by a stated construction: a mixed pairing, two copies, a filament or ribbon, or a non-abelian extension. It records that the functional is in general a real number; that integer linking and Hopf invariants, discrete flux-weighted contributions, quantised couplings and discrete operator spectra appear for different reasons and are different mathematical objects, and which one each row supplies; that what keeps it fixed differs from field to field; and what viscosity and resistivity do to it.
In plain terms
Take two closed loops of string. Count the number of times one passes through the other, with a sign for the direction. That whole number cannot be changed by any stretching or bending that does not cut a loop, and if it is not zero the loops cannot be pulled apart. (If it is zero they sometimes still cannot; the Whitehead link is the standard example, so the number is a one-way test.) Gauss wrote down, in 1833, a formula that computes this number from the shapes of the two loops: an integral over both curves that comes out, every time, as an integer. That is the linking number, and it is where this page starts.
Now replace the strings with the thin threads along which a fluid is spinning, the vortex lines, or with the lines of a magnetic field. In an ideal fluid Kelvin’s theorem, the subject of Audit 10, says the vortex lines are carried by the flow as if painted on it, and a magnetic field in a perfect conductor is carried the same way. So two linked vortex rings stay linked and two linked flux tubes stay linked, and the flow cannot change the linking number. In 1961 Jean-Jacques Moreau showed that a certain integral over the whole fluid, the velocity dotted with the vorticity, is conserved by such a flow. In 1969 Keith Moffatt, who did not know of Moreau’s note, found the conservation again, named the integral the helicity, and showed what it is measuring: for two thin linked tubes its mutual part is the linking number times the product of their strengths, times two. Lodewijk Woltjer had already shown in 1958 that the corresponding magnetic integral, the vector potential dotted with the field, is conserved in a perfect conductor, and had used it to explain why a relaxing plasma settles into a particular twisted state. Since then the same functional, or something built from it, has turned up as a measure of how a strand of DNA is coiled, as the term that gives the quantum Hall effect its quantised conductance, and as the action of a field theory whose observables are the knot polynomials of mathematics.
Three things make the audit worth doing. The first is that “the same integral” has to be earned row by row. The fluid and the plasma quantities are the same functional with different fields in it. The cross-helicity of a plasma pairs the velocity with the magnetic field, which is a related but different expression; the helicity of light adds an electric copy to the magnetic one; the DNA number is Gauss’s integral over two curves and needs a construction before it is the helicity of anything; and the knot-theory version adds a term the fluid does not have. The page states the map in each case rather than calling them all one integral. The second is that the integral and the whole number are not the same thing. For two thin tubes the mutual part of the integral is the linking number times the strengths, but a real field is not two thin tubes: it fills space, its lines need not close, and the integral is then a real number, Arnold’s average linking of every line with every other, which can take any value at all. The integer-valued examples on this page are the Gauss linking number of two closed curves and the helicity of a suitably normalised Hopf construction. A general real-valued helicity can happen to equal an integer without being quantised. Quantised circulation (Audit 13) makes the linking part of it discrete without making the whole of it an integer; the integer coupling of a properly defined quantum gauge theory multiplies the functional and does not make its value an integer; and the discrete spectrum of a quantum operator does not make its classical value, or every expectation value, an integer. These are different mathematical objects, and the page says which one each row supplies. The third is that “conserved” means different things in different rows. In an ideal fluid the helicity is conserved because the lines are carried by the flow. In free-space electromagnetism it is conserved by a symmetry between electric and magnetic fields, and only for special solutions is there also a flow carrying the lines. In a resistive plasma the ideal law no longer holds, its rate of change has no fixed sign, and whether it decays more slowly than the energy depends on the field, which is what Taylor’s hypothesis asserts for a turbulent one. And in the quantum Hall effect the functional is a term in an action, so the question of its conservation does not arise; what the topology fixes there is a coupling.
The rest of the page is a ledger. It lists the fields, what each calls the integral, when each first had it, what each assumes, and a verdict: exact when the row’s named quantity is the helicity functional of the stated field, the functional itself, or the two-curve linking integral applied to a named pair of curves, with the target named, limited when the row’s quantity reaches the functional by a stated construction, a mixed pairing, a sum of copies, a filament or ribbon model, an extension or a variation, or holds only after a named approximation, analogy when only some features line up. Every claim carries a note on how far the source was actually read.
What is being audited
Take a divergence-free vector field B on a region V whose flux 2-form is exact, so that a potential exists, B = ∇×A, and take B·n = 0 on the boundary. The helicity of B is the integral of A·B over V; in the language of forms it is the pairing of the 1-form A with its own exterior derivative. The page is built on the bilinear pairing ℬ(a, b) = ∫a∧db and its quadratic form Q(a) = ℬ(a, a), and each row says which pairing it is, with what in the slots, or by what construction it reaches one:
linking: Lk(C1, C2) = (1/4π)∮C1∮C2 (dx1 × dx2)·(x1 − x2)/|x1 − x2|³ ∈ ℤ (Gauss; ℬ for two unit-flux curves) ⇒ for N thin closed flux tubes of fluxes Φi: H = 2Σi<j Lk(Ci, Cj)ΦiΦj + Σi Hi, with Hi = (Wri + Twi)Φi² for a tube modelled as a closed ribbon (Călugăreanu; White; Fuller; Moffatt and Ricca)
general field: Q(A) = the flux-weighted average over pairs of field lines of their asymptotic linking rate; an exact theorem under its hypotheses, and any real value (Arnold)
fluid: H = Q(u♭) = ∫ u·ω dV, dH/dt = 0 for inviscid barotropic flow with conservative body forces on a volume bounded by a vortex surface, ω·n = 0, or more generally when the boundary term ∮(½|u|² − h − Φ)ω·n dS vanishes; a material volume alone is not sufficient (Moreau; Moffatt, eq. 17); mutual part ±2nΓ1Γ2 for two rings linked n times
plasma: HM = Q(A) = ∫ A·B dV, dHM/dt = 0 for ideal MHD (Woltjer); dHM/dt = −2ηm∫ B·(∇×B) dV with magnetic diffusivity ηm and no boundary term, a signed integrand; slower than the energy decay in Taylor’s turbulent regime, and not in general (Taylor’s hypothesis)
cross-helicity: HC = ℬ(u♭, A) = ∫ u·B dV, a mixed pairing, not Q of either field (Woltjer 1958b); optical: Hopt = ½[Q(A) + Q(C)] = ½∫(A·B − C·E) dV, two copies (Cameron, Barnett and Yao)
Chern–Simons: SCS = (k/4π)Q(A) locally, on a three-dimensional spacetime; for a U(1) connection on a bundle with F/2π integral, eiS is well defined on every closed oriented 3-manifold only for even k, and for any integer k once a spin structure is chosen (Belov and Moore); d(A∧dA) = F∧F, a degree-four Chern–Weil form proportional to c1² (Chern and Simons); non-abelian: (k/4π)∫tr(A∧dA + ⅔A∧A∧A), with a cubic term the abelian case lacks
Hopf: f: S³ → S², Ω the area form of S² normalised to ∫S²Ω = 1, da = f*Ω ⇒ Q(a) = ∫S³ a∧da = Hopf(f) ∈ ℤ (Whitehead; with the unnormalised area form the integral is 16π² times this)
The first line is the whole structure: a bilinear pairing of a 1-form with the exterior derivative of a 1-form, and its quadratic form, built from a field and a potential for it. Three things about it are easy to get wrong and the page tries not to. It exists only when a potential does, that is, when the flux form is exact and not merely closed (a uniform field in a periodic box has no periodic potential), and it is gauge-independent only under stated conditions, that the field be tangent to the boundary, or the region closed with the periods fixed, or a reference field supplied; without them the number depends on the choice of potential and is not an absolute observable. It is a real number: the linking-number formula holds for thin closed tubes with a stated model of each tube’s self-term, and for a general field the integral is Arnold’s flux-weighted average of asymptotic linking rates, an exact theorem under its hypotheses whose value is not restricted to integers and need not be a rational multiple of anything. And its conservation is not part of its definition: whether it is constant in time depends on the equations, the regularity of the flow, the boundary conditions and the choice of observable, which is the physics of each row. The remaining lines are the pairing in particular slots or reached by particular constructions. In the fluid rows the slots hold the velocity and its own curl, and the conservation comes from the transport of vortex lines by the flow. In the plasma rows they hold the vector potential and the magnetic field, and the conservation comes from flux freezing. The cross-helicity is the mixed pairing of the two; the optical helicity is two copies of the quadratic form. In the gauge-theory rows the same local functional is a term in an action on a three-dimensional spacetime, and what the topology fixes is a coupling, not the value of a conserved quantity, except that for a field built from a normalised map to the sphere the value itself is the Hopf invariant. In each case the row says which pairing, in which slots, by which construction; what differs is why the number stays put and whether it is a number or an integer.
Three conditions travel with the structure, and each row of the ledger is read against them.
First, the integral is a sum of ordinary closed-curve linking numbers only in the setting of thin closed tubes. There it is 2Σi<jLk ΦiΦj plus a self-term for each tube, and the self-term is the integer self-linking Wr + Tw of a closed ribbon only once the tube has been modelled as one; a tube with general internal winding contributes a flux-weighted winding that need not be that integer, and the familiar formula carries that modelling choice. For a field that fills space the integral is still defined and still conserved when the lines are frozen, and Arnold’s theorem gives it an exact meaning as the flux-weighted average of asymptotic linking rates, real-valued; Enciso, Peralta-Salas and Torres de Lizaur add that a regular integral invariant of exact divergence-free fields under all volume-preserving diffeomorphisms is a function of it. The verdict judges whether the row’s quantity is this pairing and by what construction; the dictionary column says which fields sit in the two slots and what “linking” means for them.
Second, what conserves the integral differs from row to row and is stated in the “direction” column, together with the boundary condition it needs, since a definition that is gauge-independent on a fixed volume can still receive helicity through its boundary. Transport of the lines by the flow (fluids, ideal MHD) conserves the helicity because the linking of material lines cannot change; a symmetry of the equations (duality in free-space electromagnetism) conserves the optical sum with no transport required, and for the null fields a transporting flow exists as well; where the functional is an action on a spacetime the question of its conservation as a charge does not arise; and under resistivity or viscosity the ideal law no longer follows: the rate of change is an integral with no definite sign, which can vanish, as it does for a field whose helicity is zero throughout its decay, or take either sign, and the comparison with the energy’s decay is a statement about a regime. A row is exact when its named quantity is the pairing and the row states the mechanism; it is limited when the quantity reaches the pairing by a construction, or the mechanism holds only in a limit.
Third, the integral is a real number and the linking number is an integer, and the page keeps them apart. Where the tubes’ strengths are quantised, as the circulation of a superfluid is quantised in units of h/m (Audit 13), the mutual part of the helicity of a link of singly quantised vortices is an integer multiple of (h/m)², and the self-terms, writhe and whatever twist the chosen framing supplies, are real. Where the field is the pull-back of the normalised area form of a sphere, the integral is the Hopf invariant and is an integer by construction. Where a compact gauge theory is globally defined, the integer is the coupling in front of the functional, not its value. Where the field is quantised, the optical helicity operator has a spectrum of integer multiples of ħ while its classical value and its expectation value need not be. These are different reasons for discreteness, and a row that reports a whole number has to say which one it is using.
Two columns in the ledger do the real work. The dictionary column says which two fields occupy the slots of the integral and what the integral measures for them, so that a verdict can be checked rather than believed. The direction column says what, if anything, keeps the number fixed, and under what boundary condition: transport, symmetry, continuity of a curve, or nothing. The verdict judges the correspondence within the row’s stated setting; the adequacy of the model that supplies the fields is answered on the “model” line.
The ledger
Each verdict applies to the correspondence named in that row’s dictionary column, not to the field as a whole. The physical adequacy of the model that supplies the fields is stated on the “model” line and does not enter the verdict.
| Field | Name used · earliest source located | Dictionary | Direction of implication | Assumptions added or dropped · model · where it breaks | Verdict |
|---|---|---|---|---|---|
| Mathematics | Gauss linking integral; Călugăreanu–White–Fuller theorem Lk = Wr + Tw; Whitehead integral for the Hopf invariant; asymptotic Hopf invariantGauss 1833 (Nachlass) · Hopf 1931 · Whitehead 1947 · Călugăreanu 1959, 1961 · White 1969 · Fuller 1971 · Arnold 1974 · Chern & Simons 1974 | Two closed curves ↔ the two slots of the integral, each carrying unit flux; Gauss’s double integral ↔ their linking number, an integer invariant under isotopy · a closed ribbon ↔ a curve with a framing; its self-linking Lk ↔ Wr + Tw, the writhe of the axis plus the twist of the framing, each real and neither invariant alone (Călugăreanu; White; Fuller) · a divergence-free field on a closed 3-manifold ↔ its helicity Q(A), defined when the flux class vanishes so that A exists · a map S³ → S² with the area form normalised to unit total ↔ a field whose helicity is the Hopf invariant, an integer (Whitehead) · a general divergence-free field ↔ the flux-weighted average of the asymptotic linking rates of pairs of its lines (Arnold), any real value · d(A∧dA) = F∧F ↔ the Chern–Simons 3-form as the local primitive of a degree-four Chern–Weil form, proportional to c1² for U(1), whose second Chern class is zero (Chern and Simons) | Definitions → the invariants. For closed curves and tubes the helicity is a sum of integers times fluxes; for a general field it is an asymptotic average and a real number; for the Hopf field it is an integer. Nothing physical is assumed or implied. Enciso, Peralta-Salas and Torres de Lizaur: a regular integral invariant of exact divergence-free C¹ fields on a closed 3-manifold, invariant under all volume-preserving diffeomorphisms, is a function of the helicity; a statement about that class of functionals, not about every invariant of every flow. | adds a region, a divergence-free field tangent to its boundary (or a closed manifold with trivial flux class), and a potential · model none; pure mathematicsbreaks nothing; but says nothing about whether any field’s lines are carried by anything. The writhe and twist are not separately invariant; the Hopf invariant is an integer only for fields that are pull-backs of a normalised area form; a zero linking number does not imply that two curves can be separated (the Whitehead link); the helicity of a general field is a real number and Arnold’s theorem is that it may be any real number | exact |
| Ideal fluids | Helicity; kinetic helicity; Moreau’s invariant; degree of knottednessMoreau 1961 · Moffatt 1969 · Moffatt & Ricca 1992 · Moffatt & Tsinober 1992 (review) · Kleckner & Irvine 2013 · Scheeler et al. 2014, 2017 | Velocity u and vorticity ω = ∇×u ↔ the potential and the field · H = ∫u·ω dV ↔ the helicity of the vorticity field · two thin vortex rings of circulations Γ1, Γ2 linked n times ↔ a mutual term ±2nΓ1Γ2, the total if the rings’ own self-terms vanish or are excluded (Moffatt, Sec. 1, eq. 6, p. 118; his abstract prints the value without the 2) · a single knotted or coiled tube modelled as a closed ribbon ↔ H = (Wr + Tw)Γ² (Moffatt and Ricca) · Kelvin’s transport of vortex lines (Audit 10) ↔ the reason the linking cannot change · knotted vortices made with shaped hydrofoils and their helicity measured (Kleckner and Irvine; Scheeler et al.) ↔ the laboratory statement | Euler with barotropic pressure and conservative body forces → dH/dt = 0, exactly, on a volume bounded by a vortex surface, ω·n = 0, or more generally when the helicity boundary term ∮(½|u|² − h − Φ)ω·n dS vanishes (Moreau; Moffatt, Sec. 2, eq. 17); a material volume is not by itself enough, since vorticity can cross a material boundary, and the tube formula → the integer content of the mutual terms. The direction is transport: the vortex lines are frozen in the flow, their linking is a material invariant, and the helicity is that invariant weighted by circulation. In the particular experiments of Scheeler and co-workers the measured helicity of a trefoil or a linked pair is transferred through reconnection from linking to writhe and to twist rather than lost (2014, 2017); that is what those flows did, not a universal sequence. | adds an inviscid barotropic fluid with conservative body forces (Kelvin’s conditions); closed or tangent vortex lines at the boundary · model Euler; the vortex-filament picture for the tube formulabreaks under viscosity, which for an incompressible Newtonian fluid of constant density changes helicity at a rate −2ν∫ω·∇×ω dV (in general 2∫ω·fν dV for the viscous force per unit mass fν), a signed integrand, and permits reconnection, which in the measured cases converted linking into writhe and twist before the twist dissipated; under baroclinicity, where the vortex lines are no longer frozen (Audit 10); and in two dimensions, where u·ω = 0 identically. The tube formula needs thin tubes; a vorticity field filling space has helicity but no linking number | exact |
| Superfluids and condensates | Centreline helicity; helicity of quantised vortices; superfluid helicitySalman 2017 · Kedia, Kleckner, Scheeler & Irvine 2018 · Clark di Leoni, Mininni & Brachet 2016 | A quantised vortex line with circulation nh/m (Audit 13) ↔ a thin tube of known flux · three candidate observables, which need not agree on one configuration: the centreline helicity, 2Σi<jLk ΓiΓj + Σ WriΓi², which omits any framing contribution by choice; the Seifert-framed continuum quantity, in which the phase supplies a twist through its Seifert framing and the total vanishes by cancellation between the linking-plus-writhe part and the framing’s twist contribution (Salman); and a regularised volume integral of the vorticity against a smoothed velocity, which behaves like the classical helicity at large scales (Clark di Leoni et al.) · the particle-relabelling symmetry of the Gross–Pitaevskii fluid ↔ a Noether charge that vanishes identically in the construction of Kedia et al., so that no conservation law comes from that route · the classical limit ↔ the centreline helicity of a bundle of superfluid vortices behaving like the helicity of a classical viscous flow (Kedia et al.) | Quantised circulation → the fluxes in the tube formula are integers times h/m, so the mutual-linking contribution is discrete in units of (h/m)². That does not by itself quantise the centreline helicity, because writhe can vary continuously; and a claim about another definition must be checked for that definition, since the Seifert-framed quantity vanishes by cancellation and the regularised one is a smoothed integral. No conservation theorem for the centreline or the regularised observable has been established here: the relabelling route gives none (Kedia et al.) and the Seifert-framed quantity is identically zero (Salman). The correspondence is limited: a filament observable, centreline helicity, reached by a construction and behaving like the classical quantity in the classical limit, not the classical invariant. | adds a definition of helicity for singular vorticity, by centreline (Lk + Wr), by Seifert framing, or by regularisation; each is a choice, and the choices differ · model Gross–Pitaevskii; vortex filaments for the centreline quantitybreaks at reconnection, which is the generic event for quantised vortices and changes the linking; and in the framing, on which the continuum definition depends and which the centreline quantity sets aside (Salman). In the simulations of Clark di Leoni et al. Kelvin waves carry the helicity transferred from linking into writhe | limited |
| Ideal magnetohydrodynamics | Magnetic helicity; Woltjer’s invariant; relative helicityWoltjer 1958a · Moffatt 1969, Sec. 5 · Berger & Field 1984 · Berger 1999 (review) | Vector potential A and magnetic field B = ∇×A ↔ the two slots · HM = ∫A·B dV ↔ the helicity of the magnetic field, constant in ideal MHD (Woltjer, eq. 3) · thin flux tubes of fluxes Φi ↔ HM = Σ LijΦiΦj, with self-terms Wr + Tw (Berger and Field) · a volume with B·n ≠ 0 ↔ no gauge-invariant helicity; the relative helicity HR = ∫(A + Ar)·(B − Br) dV against a reference field with the same normal component ↔ the gauge-invariant replacement (Berger and Field; the Finn–Antonsen form, which reduces to a difference of two helicities only when a mixed boundary term vanishes, as Pariat et al. spell out) · flux freezing (Alfvén; Audit 10) ↔ the reason the linking cannot change | Ideal induction equation → dHM/dt = 0, exactly, for a volume bounded by a magnetic surface or a closed domain with no boundary flux (Woltjer). The direction is transport, as for the fluid row, and the tube formula gives the integer content of the mutual terms; the vector potential’s gauge freedom is what forces the boundary condition on the definition, and the boundary flux is a separate condition on the conservation, since relative helicity on a fixed volume can be injected through its boundary. Woltjer’s second result, that the energy is stationary at fixed helicity when ∇×B = αB with α constant, is the Euler–Lagrange condition and not the conservation law; which constant-α field is the minimum depends on the domain and the flux constraints. | adds a perfectly conducting fluid; a magnetic surface or a closed domain, or a reference field for relative helicity · model ideal MHD; the flux-tube picture for the linking formulabreaks with resistivity, at the rate dHM/dt = −2ηm∫B·(∇×B) dV, a signed integrand, which permits reconnection; and on open volumes without the relative construction. It does not break in ideal Hall MHD: the Hall term is parallel to j×B and contributes nothing to the helicity rate, so magnetic helicity remains conserved there, transported by the electron flow, alongside the generalised helicity (Banerjee and Galtier); finite electron inertia is a separate case (Audit 10) | exact |
| Plasma relaxation | Taylor relaxation; Taylor’s hypothesis; selective decay; force-free statesWoltjer 1958a · Taylor 1974 · Taylor 1986 (review) · Berger 1984 · Matthaeus & Goldstein 1982 | A slightly resistive turbulent plasma ↔ a system in which the local topological invariants are destroyed and only the total magnetic helicity survives (Taylor 1986, abstract) · the total helicity ↔ approximately conserved while the energy decays, in a regime where its dissipation rate is bounded by the energy dissipation rate and a length scale (Berger 1984); not a consequence of resistivity as such, since a single force-free mode loses energy and helicity at the same fractional rate · the relaxed state ↔ Woltjer’s minimiser, ∇×B = μB with μ uniform, which for a toroidal pinch predicts field reversal (Taylor 1974) · energy, cross-helicity and magnetic helicity ↔ the “rugged” invariants of MHD turbulence, those that survive the ideal modal truncation (Matthaeus and Goldstein) | Resistive MHD → the ideal conservation law no longer follows; dHM/dt = −2ηm∫B·(∇×B) dV, with ηm the magnetic diffusivity, has no definite sign and can vanish, as for a decaying field with HM = 0 throughout (B = B0e−ηmk²tcos kz x̂, whose potential is everywhere perpendicular to it). Taylor’s hypothesis is that in a turbulent plasma with small resistivity it is conserved better than the energy: the ideal invariant of each flux tube is lost when the tubes reconnect, the total survives to a good approximation, and the energy is stationary at fixed total helicity for force-free fields with uniform μ. The correspondence is limited because the conservation is a hypothesis about a regime, with the rate of change of no fixed sign and the ordering of rates field-dependent, and because the relaxed state is selected by a variational argument whose minimiser depends on the domain, the flux constraints and the admissible modes. | adds small but finite resistivity; turbulence sufficient to reconnect; a conducting wall; the selective-decay ordering of dissipation rates · model resistive MHD; the variational principle with one constraintbreaks when the resistivity is not small, when reconnection is too slow to reach the relaxed state, when the field is not turbulent (the periodic force-free mode B = B0e−ηmk²t(cos kz, −sin kz, 0) has energy and helicity decaying together as e−2ηmk²t), and when other invariants are not destroyed. The selective decay is a statement about rates in a spectrum, not a theorem | limited |
| Cross-helicity | Cross-helicity; Woltjer’s second invariant; velocity–magnetic-field correlationWoltjer 1958b · Moffatt 1969, Sec. 5 · Matthaeus & Goldstein 1982 | Velocity 1-form u♭ and vector potential A ↔ the two slots of the mixed pairing ℬ(u♭, A), which is not Q of either field · HC = ∫u·B dV ↔ conserved in ideal barotropic MHD with conservative body force, the balance law being ∂t(u·B) + ∇·[u(u·B) + B(h + Φ − ½|u|²)] = 0 with dh = dp/ρ, so that the boundary condition is explicit (Woltjer 1958b, cited by Moffatt) · Moffatt’s reading (Sec. 5) ↔ the instantaneous mutual linkage of vortex lines with magnetic lines, a geometric representation that holds · the normalised cross-helicity ↔ the degree of Alfvénic alignment in solar-wind turbulence (Matthaeus and Goldstein) | Ideal MHD → dHC/dt = 0 under the stated boundary condition. What does not follow is the inference that both line families are material: the magnetic lines are frozen and the vortex lines in general are not, since the Lorentz force is not a gradient, so the instantaneous mutual linking is a valid representation of the number and not a statement that two families of material lines keep their linking. Limited: a mixed pairing, conserved, with the linking picture holding as a representation and not as the mechanism. | adds ideal barotropic MHD · model the same as the previous rowbreaks with viscosity or resistivity, with baroclinicity or non-conservative forces, and through the boundary flux; and in the inference, not the representation, that the linking of vortex and magnetic lines is the linking of two material families | limited |
| Dynamo theory | The α-effect; helical turbulence; kinetic and current helicitySteenbeck, Krause & Rädler 1966 · Pouquet, Frisch & Léorat 1976 · Moffatt 1978, Ch. 7 · Krause & Rädler 1980 · Brandenburg & Subramanian 2005 (review) | The mean helicity density ⟨u·ω⟩ of small-scale turbulence ↔ a statistical pseudoscalar, not an integral over a volume · the α term of the mean electromotive force, ℰ = αB̄ − ηt∇×B̄ + … in the homogeneous isotropic closure, with tensor coefficients and pumping terms otherwise ↔ the component parallel to the mean field that lack of mirror symmetry allows (Steenbeck, Krause and Rädler) · α ≈ −(τ/3)(⟨ω·u⟩ − ⟨j·b⟩/ρ) ↔ a closure-dependent estimate in the isotropic short-correlation-time approximation, kinetic minus current helicity, the second term a nonlinear correction and not part of the kinematic first-order-smoothing result (Pouquet, Frisch and Léorat; Brandenburg and Subramanian) · the inverse cascade of magnetic helicity to large scales ↔ what the conservation of the previous rows does in turbulence | Helical turbulence → an α-effect; the direction is statistical, from a correlation to a transport coefficient, and nothing is conserved or topological in it. The word is the same and the density is the same integrand, but the row’s object is a local average whose sign matters and whose magnitude is a model parameter. Analogy: the integrand of the helicity, used as a measure of handedness, in a setting where the integral’s invariance plays no role except through the inverse cascade. | adds mean-field electrodynamics; scale separation; isotropy and homogeneity for the scalar form; a closure for the turbulent electromotive force; unit conventions (μ0 = ρ0 = 1 in Brandenburg and Subramanian) · model first-order smoothing or its successors; the two-scale approximationbreaks as a closure when the scale separation fails and when the magnetic helicity constraint quenches α (the current-helicity term), which is the conservation of the ideal rows reappearing as a limit on the dynamo; the sign of α is not fixed by any topology | analogy |
| Abelian Chern–Simons theory | Chern–Simons functional; Chern–Simons term; Hopf termChern & Simons 1974 · Whitehead 1947 · Deser, Jackiw & Templeton 1982 · Polyakov 1988 · Dunne 1999 (lectures) · Tong 2016, Ch. 5 | A U(1) connection A on a 3-manifold ↔ the vector potential · ∫A∧dA ↔ the magnetic helicity, the same integral · under A → A + dχ ↔ a change by a boundary term, zero on a closed manifold for small gauge transformations (Dunne, Sec. 2; Tong, Sec. 5.1) · a large gauge transformation acting on a connection with flux ↔ a shift of a representative of SCS = (k/4π)∫A∧dA by a multiple of 2π only for suitable k, so that eiS is well defined; on a globally defined potential the shift is zero, since ∫α∧dA = 0 for closed α, and the condition arises for connections on non-trivial bundles, defined by patching or by extension to four dimensions, where eiS exists on every closed oriented 3-manifold for even k and on spin manifolds for any integer k (Tong, Sec. 5.1.3, for the torus argument; Belov and Moore, eqs 1.1–1.3, for the general statement, in a normalisation where their level is half of this k) · the Wilson loops of the abelian theory ↔ their expectation values are exponentials of Gauss linking numbers, with the self-linking needing a framing (Polyakov; Witten, Sec. 2) · a field that is the pull-back of the unit-normalised area form of S² ↔ helicity equal to the Hopf invariant (Whitehead) | Definitions → the identity of the local functionals, exactly: the Chern–Simons integrand of a U(1) connection is the magnetic helicity density of its curvature. What differs is the setting and the role. The functional is an action on a three-dimensional spacetime, not the spatial charge the other rows compare; its variation gives F = 0, so there are field equations and global and boundary degrees of freedom but no propagating bulk modes; and what a globally defined theory fixes is the coupling k, which no gauge transformation changes, together with the well-definedness of eiS. Exact for the functional, with the target named. | adds a 3-manifold; a connection on a bundle, with F/2π integral; for the quantisation of k, a compact gauge group, a closed manifold or a suitable boundary condition, and a statement of whether a spin structure is chosen · model abelian gauge theory in three dimensionsbreaks the reading “a conserved helicity”: the functional is a spacetime action, and in the quantum Hall application (next row) it is an action for a background field, not a number attached to a state. Outside the normalised pull-back case the value is not restricted to integers, as in every other row | exact |
| Quantum Hall effect | Chern–Simons effective action; Hall response; level kTong 2016, Sec. 5.1 · Deser, Jackiw & Templeton 1982 | The Chern–Simons term S/ħ = (k/4π)∫𝒜∧d𝒜 in the effective action for the background gauge field, written in the dimensionless connection 𝒜 = eA/ħ ↔ the helicity functional as a term in an action · its variation ↔ the Hall current, σxy = k e²/h; in the physical potential Tong’s coefficient carries the units, σxy = kT/2π with kT = e²ν/ħ (eq. 5.7 and the line after it, pp. 149–150), and kT = (e²/ħ)k · the level k ↔ an integer, in the integer quantum Hall setting under consideration, by the large-gauge-transformation argument on a torus (Sec. 5.1.3) · a Laughlin state ↔ an emergent gauge field with integer level m and a Hall response e²/2πħm; more generally an integer K-matrix and charge vector t with σxy = (e²/h) tᵀK⁻¹t (Tong, eqs 5.20–5.23 and 5.31), the fractional response arising from an integer internal level and not from a fractional one · a Chern–Simons term added to Maxwell theory ↔ a topologically massive photon (Deser, Jackiw and Templeton) | The functional → a response coefficient, by variation; the global definition of the theory → an integer coupling. No field lines are linked, and the integer is a coupling rather than the value of a charge; it is the Chern number of Audit 13’s Hall row seen from the effective action. The correspondence is limited: the row’s named quantity, the Hall conductance, is reached from the functional by a variation, one step removed. | adds the integer quantum Hall setting under consideration: a two-dimensional gapped system at integer filling with no further low-energy topological degrees of freedom, since the fractional states are gapped too and need the emergent field (Tong, opening of Sec. 5.2); an effective-action description; a compact gauge group · model the low-energy effective theory of a quantum Hall state; Maxwell–Chern–Simons theory for the massive photonbreaks as a helicity: the integral is of a background field over spacetime, not of a physical field over space at an instant. Integrating out the emergent field of a Laughlin state to leave a fractional coefficient for the background field is not globally valid, as Tong notes after his eq. 5.22: the fractional response comes from an integer internal level, and the level itself never becomes rational | limited |
| Non-abelian Chern–Simons and knot invariants | Chern–Simons gauge theory; Jones polynomial; Wilson-loop expectation values; framing anomalyDeser, Jackiw & Templeton 1982 · Witten 1989 · Polyakov 1988 | The non-abelian Chern–Simons form tr(A∧dA + ⅔A∧A∧A) ↔ the helicity functional with a cubic correction that the abelian case lacks · the expectation value of a Wilson loop ↔ a knot invariant, the Jones polynomial for SU(2) (Witten) · the abelian limit ↔ the Gauss linking number in the exponent (Polyakov; Witten, Sec. 2) · the dependence of a knot’s expectation value on a framing ↔ the Călugăreanu decomposition again, since the self-linking of a curve needs a ribbon · a large gauge transformation, for a stated group and trace normalisation ↔ a shift of the action by a discrete amount set by the winding of the transformation, so that eiS is well defined only for quantised k; the coupling itself is unchanged (Deser, Jackiw and Templeton, at abstract level) | The functional → knot invariants, by quantum expectation values; the abelian limit → the linking number of the mathematics row. The correspondence is limited: the objects produced are invariants of knots, not helicities of fields, the cubic term has no fluid counterpart, and the functional is again an action rather than a spatial charge. What carries over exactly is the framing: the same reason a curve has no self-linking without a ribbon. | adds a non-abelian group; a quantum field theory with the functional as its action; a framing of every loop · model Chern–Simons theory at integer levelbreaks the identification with a conserved spatial charge; the non-abelian functional shifts under large transformations, and the theory’s content is in its correlation functions, not in the value of the functional on a configuration | limited |
| Electromagnetism in vacuum | Electromagnetic helicity; optical helicity; Hopfions; knotted lightCalkin 1965 · Rañada 1989 · Trueba & Rañada 1996 · Irvine & Bouwmeester 2008 · Cameron, Barnett & Yao 2012 | The vector potential A and the field B, together with a second potential C with E = −∇×C ↔ two copies of the integral · H = ½∫(A·B − C·E) dV ↔ the optical helicity, conserved in free space (Cameron, Barnett and Yao, eqs 2.6–2.7) · the duality rotation of E into B ↔ the symmetry whose Noether charge it is (Calkin) · the difference between the numbers of right- and left-handed photons ↔ what the helicity counts, up to a factor (Trueba and Rañada) · Rañada’s null fields built from the Hopf map ↔ configurations in which every pair of magnetic lines is linked once and the magnetic helicity is the Hopf index times a constant · the linked and knotted beams of Irvine and Bouwmeester ↔ the same construction · for a null field with B ≠ 0, the velocity v = E×B/|B|² ↔ a flow with E + v×B = 0, so that ∂tB = ∇×(v×B) and the magnetic lines are transported by it where it is regular (Irvine 2010, Sec. 3) | Maxwell’s equations in vacuum → conservation of the optical helicity, the sum of two copies, exactly, by the duality symmetry; the magnetic helicity alone changes at the rate −2∫E·B dV and is not conserved in general, but is separately conserved for null fields, for which a transporting flow also exists, so that duality and transport coexist in the special solutions. For the Hopf-map fields the number is the linking of field lines; in general it is the photon-number difference. Limited: two copies of the functional, conserved by symmetry, with transport and a topological reading available for the null fields and not in general. | adds a second potential; source-free Maxwell theory · model classical electrodynamics in vacuum; the photon picture for the counting interpretationbreaks with sources or media that break the duality symmetry, where the optical helicity is not conserved; in the transporting flow where the quotient E×B/|B|² cannot be extended across the zeros of B to a regular invertible flow, which is what preserved line topology requires (for a plane wave it can be: the quotient is the constant velocity cẑ through the nodes); and in the link between the two readings, since a generic field has a well-defined helicity and no closed field lines to be linked. The Hopfion’s lines move and deform under the flow; deformation does not change their linking while the flow stays regular (Irvine and Bouwmeester study the evolution; the topological statement is Irvine 2010’s) | limited |
| Molecular biology (DNA) | Linking number; twist and writhe; supercoiling; topoisomeraseVinograd et al. 1965 · White 1969 · Fuller 1971, 1978 · Bauer, Crick & White 1980 · Koster et al. 2005 | The two strands of a closed circular duplex ↔ the two curves of Gauss’s integral · their linking number Lk ↔ an integer fixed while the backbones are intact · Lk = Tw + Wr ↔ the partition of a fixed linking between the twist of the double helix and the writhe of its axis, the supercoiling (White; Fuller; Bauer, Crick and White) · the twisted circular form of polyoma DNA ↔ the observation that started the subject (Vinograd et al.) · a topoisomerase ↔ an enzyme that cuts one or both strands and reseals them: type II enzymes pass a duplex through a double-strand break and change Lk by 2; type IA enzymes pass a strand through a single-strand break and change Lk by 1; type IB enzymes relax by controlled rotation about the intact strand and can release several turns in one cleavage event (Koster et al.) | Geometry → the invariant, exactly: the linking number of a closed ribbon is the Gauss integral, and its decomposition is the Călugăreanu–White–Fuller theorem of the mathematics row applied to a molecule. The direction is topology rather than transport: the number is fixed because the strands are continuous, and it changes only when a strand is cut. The row is exact for the invariant; no helicity integral over a field appears, because the field is two curves. | adds a closed duplex; the ribbon model of the double helix · model the elastic-rod description of DNA for how Tw and Wr partitionbreaks when a strand is cut, which is the designed exit: topoisomerases change the integer, by one, by two, or by several per event according to their class. The twist and writhe are not separately conserved and their partition is set by elasticity and by proteins | exact |
The same pairing appears in places not given rows: the relativistic helicity currents of a fluid and a plasma (Bekenstein 1987); the energy of a knotted flux tube bounded below by its helicity or crossing number under magnetic relaxation, and Moffatt’s energy spectrum of relaxed states (Moffatt 1990; Freedman and He 1991); the Hopf-map equilibria of magnetohydrodynamics (Kamchatnov 1982); and Kelvin’s vortex atoms, the best-known early proposal that knotted vortices are the constituents of matter. These are noted under extensions and not separately audited.
Historical relationships
The earliest located statement of the integral is Gauss’s note of 22 January 1833, printed in the fifth volume of his collected works among his papers on electrodynamics: the double integral over two closed curves that counts how many times one winds through the other. It was written down for a reason that belongs to this page, the work done carrying a magnetic pole around a current loop, and it sat in the Nachlass until 1867. Kelvin’s 1867 paper on vortex atoms, which proposed that atoms were knotted vortex rings in the ether and which was read in its opening pages only, is an early place, and the best known, where the knotting of vortex lines was given physical weight; Tait’s tabulation of knots followed from it, and the mathematics of linking and knotting grew up alongside a physics that then abandoned the ether and kept the knots.
The invariant itself has two beginnings, both in 1958 and 1961, and one renaming. Woltjer’s 1958 note in the Proceedings of the National Academy, which was read, states that ∫A·curlA dV is constant for a perfectly conducting fluid and proves that the field of least magnetic energy for a given value of it satisfies the force-free equation with constant α; the same volume carries his second paper introducing ∫u·B dV, which was not read and is known through Moffatt’s citation of it. Moreau’s 1961 note in the Comptes rendus, which was also not read and is known through Moffatt and Tsinober’s 1992 review, states the conservation of ∫u·ω dV for a barotropic ideal fluid. Moffatt’s 1969 paper, which was read in full, states the conservation law without citing Moreau, names the integrand the helicity per unit volume (his Section 2), and gives the topological interpretation: for two linked vortex filaments the mutual part of the integral is the linking number times the product of the circulations, with the double sum of his equation 6 (Section 1, p. 118) supplying a factor of two that his abstract omits. That this was a rediscovery rather than a borrowing is the assessment of Moffatt and Tsinober’s 1992 review, which credits Moreau with the discovery and Moffatt with the rediscovery and the interpretation; a missing citation alone would not establish it. The name and the interpretation are Moffatt’s; the theorem is Moreau’s and, for the magnetic case, Woltjer’s.
The mathematics that connects the integral to knots was mostly in place before the physics used it. Hopf’s 1931 map from the three-sphere to the two-sphere and Whitehead’s 1947 expression of its invariant as an integral of the form A∧dA are the origin of the Hopf term; Călugăreanu’s papers of 1959 and 1961 gave the decomposition of the self-linking of a closed ribbon into writhe and twist, White proved it without Călugăreanu’s curvature restriction in 1969, and Fuller named the writhing number in 1971 and applied the decomposition to DNA in 1978, after Vinograd’s group had found the supercoiled circular form of polyoma DNA in 1965. Arnold’s 1973 lectures, printed in 1974 and translated in 1986, extended the linking interpretation from closed filaments to arbitrary divergence-free fields by defining the asymptotic linking of two trajectories and showing that the helicity is its flux-weighted average, which can be any real number; Arnold and Khesin’s 1998 monograph is the standard treatment, and Enciso, Peralta-Salas and Torres de Lizaur showed in 2016 that a regular integral invariant of exact divergence-free fields under all volume-preserving diffeomorphisms is a function of the helicity. Chern and Simons’s 1974 paper on characteristic forms is the source of the name for the functional A∧dA and its non-abelian extension; Deser, Jackiw and Templeton showed in 1982 that adding it to a three-dimensional gauge theory gives the gauge field a mass and that the non-abelian coefficient must be quantised, and Witten’s 1989 paper showed that the expectation values of Wilson loops in the non-abelian theory are the Jones polynomial, with the abelian case, as Polyakov had noted in 1988, giving Gauss’s linking number.
The plasma and fluid applications ran in parallel. Taylor’s 1974 letter proposed that a slightly resistive turbulent plasma conserves its total magnetic helicity while losing its energy and so relaxes to Woltjer’s minimiser, which predicted the reversed field of the toroidal pinch; his 1986 review, whose abstract was read, states the hypothesis in its final form, that resistivity destroys all the topological invariants of the ideal plasma so that only the total helicity survives. Berger and Field in 1984 gave the gauge-invariant relative helicity for volumes through which field passes and Berger in the same year bounded its dissipation. Steenbeck, Krause and Rädler had found the α-effect of mean-field dynamo theory in 1966, and Pouquet, Frisch and Léorat in 1976 tied it to the kinetic and current helicities of the turbulence and found the inverse cascade of magnetic helicity. On the optical side Calkin in 1965 found the duality symmetry of the free Maxwell equations and its conserved charge, Rañada in 1989 built solutions of Maxwell’s equations from the Hopf map in which every pair of magnetic lines is linked, Trueba and Rañada in 1996 identified the electromagnetic helicity with the difference between right- and left-handed photon numbers, and Irvine and Bouwmeester in 2008 and Cameron, Barnett and Yao in 2012 brought the two threads into optics. In fluids, Moffatt and Ricca’s 1992 paper derived the Călugăreanu invariant from the invariance of helicity, and the experiments of Kleckner and Irvine in 2013, with the measurements of Scheeler and co-workers in 2014 and 2017, made knotted vortices and their helicity laboratory objects; Salman in 2017 and Kedia, Kleckner, Scheeler and Irvine in 2018 asked what the invariant means when the vortices are quantised. The pattern for the series, as far as the located sources show it: one pairing, written by Gauss for linking and named by Moffatt for fluids, found to be a conserved quantity of ideal transport in plasmas and fluids in 1958 and 1961, given its general meaning by Arnold in 1973, reached by mixed pairings, sums and filament constructions in the other fields, and used by gauge theory, from 1974 onward, as an action rather than a conserved charge.
Notes by row
Mathematics exact
The statement has three layers and the page keeps them separate. The first is Gauss’s integral: for two disjoint closed curves the double integral of the equation box is an integer, the linking number, and it is unchanged by any deformation that does not pass one curve through the other; a non-zero value obstructs separating the curves, while a zero value does not guarantee it, the Whitehead link being non-split with linking number zero (Gauss’s note in the Werke, vol. 5, p. 605, is known here through the standard citations and not read; the formula is textbook). The second is what happens to a single curve: it has no linking number with itself until a framing is chosen, and for a closed ribbon the linking number of its two edges splits into the writhe of the axis and the twist of the ribbon about it, Lk = Wr + Tw, each of which is a real number and neither of which is invariant on its own. Călugăreanu stated it in 1959 and 1961 under a restriction on curvature (records), White removed the restriction in 1969 (record), and Fuller named the writhing number in 1971 (abstract read). The third is what happens to a field that fills space, and it is Arnold’s: define the asymptotic linking number of two trajectories of a divergence-free field as the limit of the linking of long segments closed by short paths, average it over pairs of trajectories weighted by the flux, and the result is the helicity, which is therefore an invariant of volume-preserving diffeomorphisms; his abstract, which was read in the Collected Works reprint, describes it as the mean asymptotic rotation of the phase curves around each other and notes that it can assume any real value; the asymptotic linking of two trajectories is the linking of long closed-off segments divided by the product of the two flow times, not of their lengths (Vogel’s Definition 4, read; his Theorem 9 identifies the flux-weighted integral of it with the helicity), and the average is exact under Arnold’s hypotheses, not an approximation to a sum of integers. Arnold and Khesin’s Chapter III is the treatment (table of contents: helicity and energy in Sec. 1, the asymptotic linking number in Sec. 4). Enciso, Peralta-Salas and Torres de Lizaur (abstract read) prove that a functional on exact divergence-free C¹ fields of a closed 3-manifold, defined by a well-behaved integral kernel, is invariant under all volume-preserving diffeomorphisms if and only if it is a function of the helicity; that is a uniqueness theorem for that class of functionals, and not a claim that every invariant of every Euler flow is helicity. Two further facts belong here. Whitehead’s 1947 integral, for a field that is the pull-back under a map f: S³ → S² of the area form of S² normalised to unit total (Rudolph’s statement of the formula, read, makes the normalisation explicit; with the standard area form of total 4π the integral is 16π² times the invariant), is the Hopf invariant of the map and is an integer; it is the special case in which the helicity of a field, rather than of a pair of tubes, is a whole number. And Chern and Simons’s 1974 form is the local primitive of a degree-four Chern–Weil form, d(A∧dA) = F∧F for a U(1) connection; for a line bundle the second Chern class vanishes and F∧F/4π² represents c1², so the form is proportional to the square of the first Chern class and not to a second one, which is why the same integral is the boundary term of the integrality of Audit 11 and the action of the gauge-theory rows. The row is exact for all three layers and says nothing about which fields are carried by anything.
Ideal fluids exact
Moffatt’s 1969 paper, which was read, is the reference for the whole row. For an inviscid fluid with barotropic pressure and conservative body forces the vortex lines are frozen in the flow, and Moffatt shows that ∫u·ω dV over a volume bounded by a vorticity surface is invariant, names u·ω the helicity per unit volume, and evaluates the integral for a configuration of thin linked vortex filaments: his equation 6 (Section 1, p. 118) is the double sum Σ αijKiKj over filaments of circulation Ki with αij their linking numbers, which for two filaments gives the mutual term 2αK1K2; his abstract states the value as αK1K2, and the discrepancy is noted in the literature. For a general tube the self-term is a flux-weighted internal winding, and it equals the integer self-linking Wr + Tw only once the tube is modelled as a closed ribbon, which the familiar formula assumes. Moreau’s 1961 note (record; Moffatt and Tsinober’s review, read, credits it with the discovery and Moffatt with the rediscovery and the topological interpretation) has priority for the conservation law and is not cited by Moffatt in 1969. The conservation is a consequence of Audit 10, on a volume bounded by a vortex surface or one through whose boundary no helicity flows, the boundary term being ∮(½|u|² − h − Φ)ω·n dS (Moffatt’s equation 17, pp. 120–121): the helicity is the flux-weighted linking of vortex lines, the lines are material, and the linking of material lines cannot change. A material volume is not by itself sufficient, and the first five versions of this page said it was: the flow u = (−Ωy, Ωx, at) with Φ = ½Ω²(x² + y²) − az and constant density and pressure is an exact Euler solution with ω = (0, 0, 2Ω) and u·ω = 2Ωat, so the helicity of any material volume V₀ grows as 2ΩatV₀ with no viscosity and no singularity, because vorticity crosses its top and bottom faces (a counterexample supplied in review and checked); a fixed subvolume bounded by a vorticity surface can still receive advected helicity, so the boundary condition belongs to the conservation statement and not only to the definition. Moffatt and Ricca’s 1992 paper (abstract read) closes the gap between the integral and the Călugăreanu invariant by decomposing the helicity of a single tube into writhe and twist and deriving the invariance of Wr + Tw from the invariance of helicity. Kleckner and Irvine’s 2013 experiment (abstract read) made trefoil vortex knots and linked rings in water with shaped hydrofoils, and Scheeler and co-workers measured their helicity: in 2014 (abstract read) they found that reconnection transfers helicity from links and knots to helical coils, that is, from linking to writhe, and in 2017 (abstract read), with twist measured as well, that twist dissipates while writhe and linking are preserved and that the total is conserved even under stretching. Those experiments are in a viscous fluid, and the row’s “breaks” line is the point: viscosity changes helicity at the rate −2ν∫ω·∇×ω dV for an incompressible Newtonian fluid of constant density, and at 2∫ω·fν dV in general, whose integrand has no definite sign, and reconnection, which viscosity permits, is where in those flows the integer content of the mutual terms was converted into real-valued writhe and twist before the twist was lost; other flows need not follow that sequence. The row is exact within Kelvin’s conditions and for the tube formula within the filament picture; a vorticity field that fills space has a helicity, Arnold’s average, and no linking number.
Superfluids and condensates limited
Audit 13 established that the circulation of a quantised vortex is nh/m, so a link of vortex lines has fluxes that are integers times a fixed unit, and the mutual part of the tube formula becomes an integer times (h/m)². That is as far as the topology takes it, and the row is limited for three reasons the sources supply, all of which concern what “the helicity of a superfluid” is to mean. First, the observable has to be chosen, and the choices differ. The centreline helicity, linking plus writhe of the vortex lines, sets aside any framing contribution by construction. Salman (2017, abstract and Sections 3–4 read at the reviewer’s pinpoints) shows that the phase of the order parameter nevertheless supplies a twist, through the framing given by a Seifert surface of the vortex, and that when this framing is used the continuum helicity vanishes identically, the mutual-linking and writhe contributions cancelling against the framing’s twist contribution (two linked planar rings show that the linking has to be in the cancellation: their writhes are zero and their mutual term is not); the natural spanwise direction points along a surface of constant velocity potential, and he concludes that the Gauss linking number is the more appropriate quantity. The vanishing is a cancellation, not an absence of twist, and the first draft’s statement that a quantised vortex has no twist to define is withdrawn. Third, a regularised volume integral, of the vorticity against a smoothed velocity, is what Clark di Leoni, Mininni and Brachet (2016, full text read) compute in Gross–Pitaevskii simulations of reconnecting knots (their equation 8); it reproduces the classical values at large scales and shows the helicity passing into Kelvin waves. Second, the sources used here establish no general non-trivial conservation law for the centreline or the regularised observable through the evolution being discussed; the Seifert-framed quantity is identically zero wherever the construction applies, which is invariance of a trivial kind. Kedia, Kleckner, Scheeler and Irvine (2018, abstract and Sections III–IV read) look for one through the particle-relabelling symmetry that gives the classical law, in a Thomas–Fermi treatment, and find that the Euler and phase contributions cancel so that the Noether charge vanishes identically (their equation 4); that closes the relabelling route, and it is not a proof that no continuum helicity could be conserved by another construction. What they recover is a classical limit, in which the centreline helicity of a bundle of superfluid vortices behaves like the helicity of a classical viscous flow. So the superfluid has the quantised fluxes that would make the mutual terms discrete, several candidate observables that need not agree on one configuration, and no established non-trivial invariance theorem for the centreline or regularised one: the row is where Audit 13’s integer meets this page’s real number, and the answer depends on which observable is chosen. What a hydrodynamic theory built on quantised vortices owes this row is stated in the note below the verdict.
Ideal magnetohydrodynamics exact
Woltjer’s 1958 note, which was read, is three pages long and contains the row. For a perfectly conducting fluid the integral ∫A·curlA dV is constant (his equation 3), and the field that minimises the magnetic energy for a given value of it satisfies curlH = αH with constant α (his equation 12); his theorem is that the constrained minimum satisfies that equation, not that every constant-α field is a minimum: curl eigenfields with different eigenvalues can share a value of the helicity while their energies αHM/2μ0 differ. The conservation is flux freezing, Alfvén’s theorem of Audit 10, read through Gauss: the magnetic lines are material, their linking is invariant, and the helicity is that linking weighted by flux, Σ LijΦiΦj for thin tubes, with the self-terms again Wr + Tw. Two conditions are essential. The integral changes by ∮χB·n dS under A → A + ∇χ, so it is an absolute, gauge-independent quantity for every regular χ only on a volume bounded by a magnetic surface or on a closed domain, with the periods of the potential fixed where the domain has non-contractible loops; for a volume through which field passes Berger and Field (1984, abstract read) showed that there is no absolute helicity and constructed the relative helicity. Its gauge-independent form, due to Finn and Antonsen, is HR = ∫(A + Ar)·(B − Br) dV with (B − Br)·n = 0 on the boundary, invariant under independent regular gauge changes of the two potentials; expanding it gives the difference of the two helicities plus a mixed term, ∮(A×Ar)·dS, and the first draft’s “difference of two helicities” holds only when that term vanishes (Pariat et al. 2015, Section 2.2, equations 10–14, read). Gauge independence is a property of the definition; conservation needs the boundary flux as well, and relative helicity on a fixed volume can be injected through the boundary by ideal motions. The resistive rate dHM/dt = −2ηm∫B·(∇×B) dV, with ηm = ηe/μ0 the magnetic diffusivity and no boundary contribution (textbook; Berger 1999 at abstract level), has an integrand of no definite sign, unlike the Joule heating, and is what the next row makes use of. Moffatt’s 1969 paper, Section 5, treats the magnetic case alongside the fluid one and reads ∫A·B dV as the linkage of the magnetic lines. The row is exact for ideal MHD on a magnetic surface. The first draft said that the Hall regime replaces magnetic helicity with a different invariant, and that was wrong: in ideal Hall MHD the Hall term is parallel to j×B, so its contribution to the helicity rate vanishes, and with a barotropic electron pressure the magnetic helicity is conserved, transported by the electron flow, alongside the generalised helicity of the canonical momentum (Banerjee and Galtier 2016, abstract read); that is consistent with Audit 10’s statement that the Hall term transfers the freezing to the electrons. Finite electron inertia and other pressure closures are separate cases.
Plasma relaxation limited
Taylor’s argument is what a conservation law does when it fails slowly. In an ideal plasma every flux tube has its own helicity and all of them are invariant, which constrains the field so strongly that it cannot relax; with a little resistivity the tubes reconnect and the individual invariants are lost. Taylor’s hypothesis, stated in his 1974 letter (abstract read: with small departures from perfect conductivity “topological constraints are relaxed and the final state becomes unique”) and in final form in his 1986 review (abstract read: resistivity destroys all the topological invariants “so that only total magnetic helicity survives”), is that the total helicity is conserved to a good approximation while the energy decays, so that the plasma relaxes to Woltjer’s minimiser, ∇×B = μB with μ uniform over the whole volume rather than constant along each line; for the toroidal pinch that predicted the reversed toroidal field that had been seen and not explained. The correspondence is limited because the conservation is approximate and the approximation is a statement about rates in a particular regime. It is not a consequence of resistivity by itself: for the periodic force-free field B = B0e−ηmk²t(cos kz, −sin kz, 0), with ∇×B = kB, vanishing Lorentz force and the periodic potential A = B/k, the energy and the helicity both decay as e−2ηmk²t, at the same fractional rate. Berger’s 1984 paper (record; its result is known through Pariat and co-workers’ 2015 account) bounds the helicity dissipation by the energy dissipation and a length scale, so that in the corona the helicity decay time far exceeds the energy decay time; Matthaeus and Goldstein (abstract read) call energy, cross-helicity and magnetic helicity the rugged invariants of MHD turbulence, meaning those that survive the ideal modal truncation, and measured them in the solar wind; slow dissipative decay is a separate claim. What the row keeps from the exact rows is the integral and the variational principle; what it drops is the theorem. The relaxed state is selected by making the energy stationary subject to one constraint, which gives the Euler–Lagrange condition ∇×B = μB; which constant-μ field is the least-energy state depends on the domain, the flux constraints and the admissible modes, and whether one constraint is the right number, rather than the whole family of ideal invariants or none of them, is decided by how fast reconnection proceeds relative to dissipation, which is dynamics.
Cross-helicity limited
The cross-helicity ∫u·B dV is the mixed pairing ℬ(u♭, A), the velocity 1-form paired with the exterior derivative of the vector potential, and not the quadratic form of either field; it is conserved in ideal barotropic MHD with a conservative body force, and Woltjer’s second 1958 paper is the source (record; Moffatt’s 1969 Section 5, read, attributes the invariant to it). The balance law makes the hypotheses explicit: with dh = dp/ρ and body force −∇Φ, ∂t(u·B) + ∇·[u(u·B) + B(h + Φ − ½|u|²)] = 0, so the Lorentz-force contribution cancels in the volume integral and what remains is a boundary flux; calling the quantity a Casimir of the ideal bracket is another description of the same conservation, not a substitute for those hypotheses. Moffatt reads the integral as the mutual linkage of vortex lines with magnetic lines, and as an instantaneous geometric representation that reading is correct. What does not follow is the further inference that the constancy expresses the invariance of a linking of two material families: in ideal MHD the magnetic lines are frozen and the vortex lines in general are not, since the Lorentz force is not a gradient and Kelvin’s theorem fails for the fluid, which is Audit 10’s point about which velocity does the carrying. The row is limited on two counts, then: the pairing is mixed rather than the functional itself, and the linking picture holds as a representation of the number and not as the mechanism of its conservation. In turbulence the normalised cross-helicity measures the alignment of velocity and magnetic fluctuations, the Alfvénic state, and Matthaeus and Goldstein (abstract read) treat it with the magnetic helicity as a rugged invariant; that use is statistical and belongs with the dynamo row.
Dynamo theory analogy
The row is included because “helicity” in dynamo theory is the same integrand and a different object. Steenbeck, Krause and Rädler’s 1966 paper (abstract read) found that turbulence lacking mirror symmetry, in their derivation a rotating stratified medium acted on by the Coriolis force, produces a mean electromotive force with a component parallel to the mean magnetic field, the α-effect, which is what a mean-field dynamo needs; their abstract frames the result through the Coriolis force and inhomogeneity and not through helicity. The relation ⟨u×b⟩ = αB̄ is only the leading term: the mean electromotive force in a homogeneous isotropic closure is ℰ = αB̄ − ηt∇×B̄ + …, with a turbulent diffusion term, and in anisotropic or inhomogeneous turbulence the coefficients are tensors and pumping terms appear (Brandenburg and Subramanian, equations 10.48–10.49, p. 185, located in review and not reached by me, in units μ0 = ρ0 = 1). The scalar estimate α ≈ −(τ/3)⟨u·ω⟩ is the isotropic short-correlation-time closure (Moffatt 1978, Chapter 7, at record level; Krause and Rädler, record), and the current-helicity correction, α ∝ −(⟨ω·u⟩ − ⟨j·b⟩/ρ), is a nonlinear result and not part of the kinematic first-order-smoothing formula (Pouquet, Frisch and Léorat, record; Brandenburg and Subramanian, equation 9.13, p. 152, located in review and not reached by me), who also found the inverse cascade of magnetic helicity to large scales. What is used here is the helicity density as a statistical pseudoscalar, a local average that enters α with a sign, kinetic helicity minus current helicity in the nonlinear form, so that the sign of the kinetic helicity alone does not fix the sign of α, and whose magnitude is a model parameter; nothing is integrated over a volume with a boundary condition, nothing is conserved, and no lines are linked. The one place the invariance of the previous rows enters is as a constraint, and only under assumptions, with changes distinguished from values: with no boundary flux and negligible resistive change of the total, transfer produces opposite changes in the large-scale and small-scale contributions, ΔHL = −ΔHs; the contributions themselves are approximately equal and opposite only if the initial total is approximately zero, and a closed boundary alone does not remove the resistive change (Brandenburg and Subramanian, equations 9.11–9.12, p. 151, located in review and not reached by me). The current-helicity term in α is that small-scale contribution quenching the dynamo. The verdict is analogy: the same word and the same integrand, used as a closure-dependent transport coefficient rather than a universal conversion of kinetic helicity into α, in a closure whose validity is a separate question.
Abelian Chern–Simons theory exact
The identity is exact at the level of the local functional and the setting and role are different, and the row states all three. For a U(1) connection A on a three-manifold, ∫A∧dA is, in components, ∫A·(∇×A) dV: the Chern–Simons integrand is the magnetic helicity density of the curvature. Chern and Simons’s 1974 paper (record) gives the form and the fact that its exterior derivative is F∧F, a degree-four Chern–Weil form; for a line bundle this represents a multiple of c1², the second Chern class being zero, so that with F/2π of integral periods, (1/4π²)∫XF∧F = ⟨c1², [X]⟩ is an integer on a closed four-cycle X, the integer of Audit 11. On a four-manifold W with boundary the same integral need not be an integer, and where F = dA for a global potential on W it is Stokes’ theorem that ties the two: (1/4π²)∫WF∧F = (1/4π²)Q(A|∂W), the helicity of the restriction. For non-trivial bundles there is no such global potential and the patching or extension construction below is what applies. Under a gauge transformation A → A + dχ the functional changes by a total derivative, ∫d(χ dA), which vanishes on a closed manifold for a transformation connected to the identity (Dunne’s lectures, Section 2, read; Tong’s notes, Section 5.1, read, p. 149); that is the boundary condition of the MHD row in the language of forms. The first draft then said that a large gauge transformation shifts the action by 2π times k times an integer and that the level changes in integer steps, and both statements need repair. The level k is a coupling of the theory; a gauge transformation acts on the connection and cannot change it. And for a globally defined potential on a closed manifold a large transformation adds a closed 1-form α, under which Q(A + α) − Q(A) = ∫α∧dA = −∫d(α∧A) = 0, so the sector in which magnetic helicity is defined produces no shift at all. The shift, and with it the condition on k, arises for connections on non-trivial bundles, which have no single global potential and must be handled by patching or by extending the connection to a four-manifold; then a representative of the action can change by a multiple of 2π while eiS stays fixed, provided k is suitably quantised. The precise statement depends on the setting: with F/2π integral and the action (k/4π)∫A∧dA, eiS is well defined on every closed oriented three-manifold only for even k, and once a spin structure is chosen for any integer k, because the change between two four-dimensional extensions is kπ times the self-intersection ⟨c1², [X]⟩, which is even on a closed spin four-manifold (Belov and Moore 2005, equations 1.1–1.3, read; they normalise the curvature to integral periods, so their level is half of this k, and they describe the spin case as allowing half-integral values). Tong’s torus argument in Section 5.1.3 (read, p. 154) is the integer-k case of this, and Dunne’s Section 2.6 (read) gives the non-abelian version, where the shift is 8π²κN for a transformation of winding number N and forces κ to be an integer over 4π (his equations 58–61); the first draft cited that section for the abelian argument, which it does not contain. The functional is an action on a three-dimensional spacetime, so the question of its conservation as a spatial charge does not arise; varying it gives F = 0, which is a field equation, and the theory has global and boundary degrees of freedom without propagating bulk modes. Two further connections close the row. Polyakov’s 1988 letter (abstract read; the linking argument is known through secondary accounts and Witten’s exposition) showed that the expectation value of a pair of Wilson loops in the abelian theory is an exponential of their Gauss linking number, with the self-linking of a single loop needing a framing, which is the Călugăreanu decomposition returning as an ambiguity in a quantum field theory. And Whitehead’s integral, for a connection whose curvature is the pull-back of the unit-normalised area form of S², is the Hopf invariant, the one case on this page in which the value of the functional on a field, not on a pair of tubes, is an integer by construction. The row is exact for the local functional, with its target named; the next two rows are what physics does with it.
Quantum Hall effect limited
Tong’s Chapter 5 (read) is the account followed. In the integer quantum Hall setting under consideration, a gapped two-dimensional electron system at integer filling with no further low-energy topological degrees of freedom, the response to a background electromagnetic field is captured by an effective action whose leading term is the Chern–Simons functional of the background field. Written in the dimensionless connection 𝒜 = eAphys/ħ it is S/ħ = (k/4π)∫𝒜∧d𝒜, and varying it gives a current proportional to the electric field and perpendicular to it, the Hall current, with σxy = k e²/h. Tong writes the same term in the physical potential and has not set e = ħ = 1 at this point: his coefficient kT carries the units, σxy = kT/2π with kT = e²ν/ħ (his equation 5.7 and the line after it, pp. 149–150), so the translation is kT = (e²/ħ)k, and the level compared across this page is the dimensionless k. The large-gauge-transformation argument on a torus (his Section 5.1.3) requires k to be an integer in the integer quantum Hall setting under consideration, which is the integer of Audit 13’s Hall row seen from the action rather than from the band’s Chern number. The fractional states are also gapped, and for them the single-term model is wrong: a Laughlin state at filling 1/m is described by an emergent gauge field a coupled to the background, with an integer level m for a and a mixing term (Tong’s equation 5.20), and the Hall response comes out as e²/2πħm (his equation 5.23); more generally an abelian state has an integer K-matrix and a charge vector t, with σxy = (e²/h) tᵀK⁻¹t (his equation 5.31). The first draft said the integer “becomes a rational” in the fractional case, and that conflated the internal level, which stays an integer, with the response coefficient, which is a rational built from it; Tong warns after his equation 5.22 that integrating the emergent field out to leave a fractional coefficient for the background field is not globally valid, since it loses the flux restrictions that the compact emergent field carries. Deser, Jackiw and Templeton’s 1982 papers (abstracts read) are the origin of the Chern–Simons term as a term in a three-dimensional gauge theory, where it gives the photon a mass and, in the non-abelian case, must have a quantised coefficient. The row is limited because the row’s named quantity, the Hall conductance, is reached from the functional by a variation: the functional is a term in an action for a background field over spacetime, not a number attached to a field configuration at an instant, and the integer in front of it is a coupling. No field lines are linked, and the integer is a coupling rather than the value of a charge; the response current itself satisfies a continuity equation, so the first draft’s “no spatial charge is conserved” was too broad and is withdrawn. The relation to the rest of the page is exact at the level of the functional and one step removed at the level of physics, which is what limited means on this page.
Non-abelian Chern–Simons and knot invariants limited
Witten’s 1989 paper (abstract read) showed that the quantum field theory whose action is the non-abelian Chern–Simons functional, tr(A∧dA + ⅔A∧A∧A), has as the expectation values of its Wilson loops the Jones polynomial and its relatives, so that a functional built from the helicity integral produces the knot invariants of mathematics. Three things connect it to this page and three things separate it, and the verdict weighs them. Connecting: the abelian limit of the theory gives, as Polyakov had shown (abstract read; the argument is known through Witten’s Section 2 and secondary accounts), the Gauss linking number in the exponent of a two-loop expectation value; the expectation value of a single loop depends on a framing of the loop, for the same reason a single curve has no self-linking without a ribbon, so the Călugăreanu decomposition governs the quantum theory as it governs a vortex tube; and the level must be an integer because the action changes under large gauge transformations by a discrete amount fixed by the winding of the transformation (Deser, Jackiw and Templeton, abstracts read; Dunne, Section 2.6, read: equations 58–61, the shift S → S − 8π²κN under a transformation of winding number N, so that κ is an integer over 4π; his equation 63 gives d tr(A∧dA + ⅔A∧A∧A) = tr(F∧F), the non-abelian counterpart of d(A∧dA) = F∧F). Separating: the cubic term has no counterpart in any fluid or plasma helicity, since the fields there are abelian; the objects the theory produces are invariants of knots, polynomials, and not helicities of fields; and the functional is again an action, so that the content of the theory is in its correlation functions and not in the value of the functional on a configuration, which is defined only modulo the large transformations. The row is limited: the same functional, generalised, in a role that turns the linking number into a family of finer invariants and leaves the fluid’s conserved quantity behind.
Electromagnetism in vacuum limited
The source-free Maxwell equations conserve a helicity, and the row is limited because the quantity is two copies of the functional and because the mechanism, in general, is a symmetry rather than transport. Calkin’s 1965 paper (abstract read) found that the free equations are invariant under a rotation of E into B, the duality rotation, and that the conserved quantity this symmetry generates is proportional to the difference between the numbers of right- and left-circularly polarised photons; Cameron, Barnett and Yao (2012, full text read) write it as H = ½∫(A·B − C·E) dV with C a second potential, E = −∇×C (their equation 2.6, in units with ε0 = μ0 = c = 1), show dH/dt = 0 (their equation 2.7), and give the quantised operator as ħΣ(n̂L − n̂R) summed over modes (their equation 2.8), whose eigenvalues are integer multiples of ħ while the classical value and the quantum expectation value need not be; they cite Candlin rather than Calkin for the duality connection. Trueba and Rañada (1996, abstract read) had identified the same sum of magnetic and electric helicities as a constant of the motion equal to twice the classical limit of the photon-number difference. So the quantity is ½[Q(A) + Q(C)], and its conservation is the Noether charge of duality. The magnetic helicity on its own changes at the rate −2∫E·B dV when there is no boundary contribution, and is therefore not conserved in general; it is conserved for null fields, on which E·B = 0. The first draft said that the field lines of a radiating field are carried by no flow, and that is false for exactly the fields the row goes on to discuss: wherever E·B = 0 and B ≠ 0, the velocity v = E×B/|B|² satisfies E + v×B = 0, so Faraday’s law reads ∂tB = ∇×(v×B), the flux-transport equation of Audit 10, and for a null field the electric lines are carried by the same flow; Irvine (2010, Section 3, read at the reviewer’s pinpoint; the paper has a 2017 corrigendum not inspected) makes this the basis of a helicity-conserving flow for null fields. Preserved line topology needs the flow to be regular and invertible, which fails where the quotient E×B/|B|² cannot be extended to a regular flow across the zeros of B; a zero is not by itself fatal, since for a plane wave the quotient is the constant velocity cẑ and extends through the nodes, but in general the extension has to be shown. The topological reading therefore survives for special fields on two grounds. Rañada’s 1989 construction (abstract read) builds null solutions from the Hopf map in which every pair of magnetic lines is linked once and the magnetic helicity is the Hopf index times a constant, and Irvine and Bouwmeester’s 2008 paper (abstract read) presents the same fields as linked and knotted beams of light and studies their evolution; the lines move and deform, and deformation does not change a linking number while the flow stays regular, so the first draft’s “preserved for a time and then the lines deform” conflated two different things. For a generic field there are no closed field lines to be linked and the integral counts photons of each handedness instead. Limited: two copies of the functional, conserved by symmetry, with transport and a topological reading available for the null fields and not in general.
Molecular biology (DNA) exact
The row is the mathematics row applied to a molecule, with nothing transported and the integer protected by chemistry. Vinograd and co-workers found in 1965 (record) that the closed circular DNA of polyoma virus is twisted into a supercoil; White (1969, record) and Fuller (1971, abstract read; 1978, abstract read) supplied the decomposition, and Bauer, Crick and White’s 1980 article (record; the publisher’s page gives a start page of 118 and DOI 10.1038/scientificamerican0780-118 while the PubMed record gives 100–113, a conflict not resolved here) made it standard: the two strands of a closed duplex are two closed curves, their linking number is Gauss’s integer, and while both backbones are intact it cannot change, so that any change in the twist of the double helix is compensated by a change in the writhe of its axis, Lk = Tw + Wr. The row is exact for the invariant: the linking number of the strands is the Gauss integral and its decomposition is the Călugăreanu–White–Fuller theorem, with no approximation. What the row lacks is the field: there is no helicity integral over a volume, because the object is two curves and not a distribution of flux, so the integer is the linking number itself and not a flux-weighted average. The exit from the invariant is the designed one, and the first draft described it too uniformly: type II topoisomerases pass a duplex through a transient double-strand break and change Lk by two, type IA enzymes pass a strand through a single-strand break and change it by one, and type IB enzymes relax supercoils by a controlled rotation about the intact strand, releasing several turns in a single cleavage event, with the number per step rising with the torque (Koster et al. 2005, abstract read); each is an enzymatic version of reconnection, and the step size belongs to the enzyme class and not to the topology. Fuller’s 1978 paper (abstract read) treats the decomposition of the linking number of a ribbon as a problem posed by DNA wound on nucleosomes, and the partition of a fixed Lk between Tw and Wr is set by the elasticity of the molecule and by the proteins bound to it, which is the model line and not the topology.
The dictionary
Terms that name the same object across rows, within each row’s setting, except where an entry says otherwise.
- The pairing
- ℬ(a, b) = ∫a∧db, and its quadratic form Q(a) = ℬ(a, a) = ∫A·B dV for a divergence-free B = ∇×A with an exact flux form and B·n = 0 · Q(u♭): helicity (fluids) · Q(A): magnetic helicity (MHD), and the local Chern–Simons functional (gauge theory) · ℬ(u♭, A): cross-helicity, a mixed pairing · ½[Q(A) + Q(C)]: optical helicity, two copies · ℬ for two unit-flux curves: the Gauss linking number (DNA) · linking plus writhe of filaments with quantised flux: centreline helicity (superfluids), a construction · tr(A∧dA + ⅔A∧A∧A): the non-abelian extension, with a cubic term · one pairing, reached in six different ways
- The two slots
- a 1-form and the exterior derivative of a 1-form: the velocity 1-form and its own derivative, the vorticity · the vector potential and its own derivative, the magnetic field · the velocity 1-form and the magnetic field, which is the mixed case · a connection and its curvature · two unit-flux curves · which pair sits in the slots is the first thing a row has to say
- What it measures
- for thin closed tubes, 2Σi<jLk ΦiΦj plus a self-term for each tube, the self-term being the integer Wr + Tw only for a tube modelled as a closed ribbon · for a field filling space, Arnold’s flux-weighted average of asymptotic linking rates, exact under its hypotheses and any real number · for a pull-back of the unit-normalised area form of S², the Hopf invariant · for a radiating electromagnetic field, the difference between right- and left-handed photon numbers · for a Chern–Simons action, nothing about a state; its coefficient is a coupling, and in the Hall application the response follows from it by variation, through the emergent-field construction where the state requires one
- What keeps it fixed, and under what boundary condition
- transport of the lines by the flow, on a volume bounded by a vortex or magnetic surface or with no helicity flux through the boundary (ideal fluids; ideal MHD, including ideal Hall MHD, on a fixed perfectly conducting boundary with v·n = 0 and B·n = 0) · the balance law of the mixed pairing, with the Lorentz-force term cancelling and a boundary flux remaining (cross-helicity) · the duality symmetry (vacuum electromagnetism), with a transporting flow as well for null fields · continuity of the strands (DNA) · no longer the ideal theorem, which does not apply in general; the balance law determines the change, which has no fixed sign, and Taylor’s ordering of the helicity and energy decay rates is a conditional hypothesis for its turbulent plasma regime, failing for a single force-free mode (resistive plasmas; viscous fluids) · not applicable, the functional being an action and not a charge (Chern–Simons in the Hall effect and in knot theory)
- The integer, and its reason
- the linking number of two closed curves, or of DNA strands: closed-curve topology · the self-linking of a closed ribbon: a specified framing · the Hopf invariant: a normalised pull-back · the mutual-linking part of a superfluid link’s centreline helicity, an integer times (h/m)²: quantised flux, with the writhe continuous · the coupling k of a globally defined compact Chern–Simons theory, even or any integer according to the spin structure: a property of the theory, unchanged by any gauge transformation · the eigenvalues of the optical helicity operator, integer multiples of ħ: quantisation of the field · not the helicity of a general field, which is a real number · not the writhe or the twist, which are real and not separately invariant
- The framing
- the ribbon a single curve needs before it has a self-linking · the twist of a vortex or flux tube about its axis · the Seifert framing that the phase of a superfluid supplies, which the centreline helicity sets aside and the continuum helicity includes, to vanish by cancellation (Salman) · the framing of a Wilson loop, on which its expectation value depends (Witten; Polyakov) · one problem, met four times
- How the invariant changes
- by reconnection, which in the measured classical flows converted linking into writhe and twist before the twist dissipated, and which is the generic event for quantised vortices · by cutting a strand, in steps of one, two or several according to the enzyme (DNA) · by the failure of Kelvin’s conditions, viscosity or baroclinicity (fluids), by the boundary flux on a fixed volume (fluids; relative helicity), or by sources or media that break the duality symmetry (electromagnetism) · not at all where the functional is an action, and the coupling of such a theory is not changed by any gauge transformation
- Chern–Simons form
- A∧dA for U(1), the integrand of the quadratic form · tr(A∧dA + ⅔A∧A∧A) for a non-abelian group, with the cubic term that no fluid or plasma helicity has · its exterior derivative, d(A∧dA) = F∧F for U(1), a degree-four Chern–Weil form representing a multiple of c1² with (1/4π²)∫XF∧F = ⟨c1², [X]⟩ an integer on a closed four-cycle when F/2π has integral periods, Audit 11’s integer, while on a four-manifold W with boundary the integral need not be an integer and, for a global potential with F = dA, (1/4π²)∫WF∧F = (1/4π²)Q(A|∂W) by Stokes’ theorem, which is the sense in which the helicity is the boundary term · for the non-abelian form, d tr(A∧dA + ⅔A∧A∧A) = tr(F∧F) (Dunne, Sec. 2.6, eq. 63)
Verdict
Three claims are usually run together. The ledger supports the first only once the map from each field to the functional has been stated, supports the second only after the mechanism and its boundary condition have been named row by row, and rejects the third except in the cases it lists.
First: a common helicity functional, a potential paired with its own flux, appears exactly in the helicity of an ideal fluid, the magnetic helicity of a plasma and the local abelian Chern–Simons expression, and for thin closed tubes it is twice the flux-weighted sum of the mutual linking numbers plus a self-term for each tube. The other rows reach it by a stated construction: a mixed pairing of velocity with magnetic field for the cross-helicity, two copies for the optical helicity, a filament construction with quantised flux for the superfluid, the two-curve case of the pairing for a DNA duplex, a variation for the Hall response, and a non-abelian extension with a cubic term for knot theory. These connections are precise, and they are not all literal substitutions into one quadratic integral. Five of the twelve rows are exact: the mathematics, the ideal fluid, ideal magnetohydrodynamics, the abelian Chern–Simons functional and the DNA duplex, each with its target named. Six are limited: the superfluid, because the observable has to be constructed and no non-trivial conservation theorem for the centreline or regularised observable is established; plasma relaxation, because the conservation is a hypothesis about a regime; the cross-helicity, because the pairing is mixed and the linking picture is a representation rather than the mechanism; the quantum Hall effect, because the row’s quantity is reached from the functional by a variation; non-abelian Chern–Simons theory, because of the cubic extension; and vacuum electromagnetism, because the quantity is two copies conserved by a symmetry. One is an analogy, the α-effect, which uses the integrand as a closure-dependent transport coefficient.
Second: defining the quantity does not establish its conservation. The equations, the regularity of the flow, the boundary conditions and the choice of observable decide whether it stays fixed, and the page states each. Transport of material lines conserves it in ideal fluids and ideal plasmas, on a volume bounded by a vortex or magnetic surface or one with no helicity flux through its boundary, a material volume alone not sufficing, and that is the only mechanism under which “the linking cannot change” is the reason. The duality symmetry conserves the optical sum, and for null fields a transporting flow exists as well, so the two routes can coincide in special solutions. The continuity of two strands conserves the DNA number. In a resistive plasma the ideal law no longer applies and the rate of change has no fixed sign; Taylor’s hypothesis is a regime-dependent ordering of rates, proposed for suitable relaxing regimes of a turbulent plasma with small resistivity and false for a single decaying force-free mode. Where the functional is an action its role must be distinguished from a conserved charge. The topology establishes invariance under a specified class of deformations; the dynamics, or the symmetry, or the chemistry, decides whether the evolution stays within that class, and “helicity is conserved because it is topological” skips that second step.
Third: that the helicity is a topological integer. It is in general a real number. For thin tubes the mutual terms are integers times products of fluxes, and the fluxes are real; for a general field it is Arnold’s average, which can take any value; for a superfluid link the mutual-linking part is discrete in units of (h/m)² while the writhe varies continuously; and a viscous or resistive fluid can convert the integer content of a link, through reconnection, into writhe and twist. Integer results need additional structure, and the reasons differ: closed-curve linking, where the fluxes are one; a specified framing, for a self-linking; a normalised Hopf pull-back, for a field; quantised flux, for the mutual terms of a superfluid link; the coupling of a globally defined compact gauge theory, which multiplies the functional rather than being its value and which no gauge transformation changes; and the spectrum of the quantised optical helicity operator, which is discrete where its classical value is not. Audit 13 supplied the integer periods of a phase; this page is where they meet a functional that is a number, and the ledger records, row by row, what has to be added before it is anything else.
What this means for a hydrodynamic theory of quantum mechanics or gravity. Kelvin’s vortex atoms are the earliest well-known proposal to make particles out of knotted or linked vortices, and this page is the ledger any such theory has to settle. It has four questions to answer. Which observable is defined: the centreline helicity, which sets the framing aside; the Seifert-framed continuum quantity, which with the natural framing vanishes by cancellation (Salman); or a regularised volume integral; these need not agree, and where the vortices are singular the definition must say how the singularity is regularised or framed. On which fields and which domain, with which potential and which boundary condition, since the functional is not an absolute quantity without them. Which equations and boundary conditions preserve it, given that transport conserves a helicity on this page only where the lines are material, that the relabelling route in the Gross–Pitaevskii fluid yields no conserved charge (Kedia and co-workers), and that reconnection, which is generic for quantised vortices, changes the linking; the burden on the theory is to define its observable and prove its invariance, not to overcome a theorem that excludes every possible definition, since no such theorem is on this page. And which additional structure supplies the claimed discreteness and its physical unit: with circulation nh/m the mutual-linking part of a link’s helicity is discrete in units of (h/m)², but the writhe varies continuously and the twist depends on the framing, so a theory that identifies a particle number or a charge with the centreline helicity of a tangle has identified it with a quantity that changes continuously under deformation of the tangle’s axes; a theory that uses another observable has to say which, and show what it does. If the theory reaches for the Chern–Simons functional as a way of making the integer appear, the integer there is the coupling of a globally defined compact gauge theory, fixed by the requirement that eiS exist and unchanged by any gauge transformation, and it multiplies an action; it is not the value of a fluid observable. What this page does give such a theory is the exact statement it can build on: in an ideal fluid the flux-weighted linking of vortex lines is invariant on a volume bounded by a vortex surface, and where the fluxes are quantised the mutual terms are discrete. The rest is the self-terms, the framing, the regularisation, the boundary and the reconnection, and the ledger is a list of the fields that have had to deal with each.
What this page does not claim
It does not claim that every quantity in the ledger is literally one quadratic integral; the fluid, plasma and abelian Chern–Simons quantities are, and the others reach it by the constructions the rows state. It does not claim that helicity is an integer, or that it is a topological invariant in the sense of taking discrete values; it is an integral invariant of volume-preserving transport, real-valued, and Arnold’s theorem is that it may be any real number. It does not claim that a zero linking number means two curves can be separated. It does not claim that the tube formula’s self-terms are integers for tubes that have not been modelled as closed ribbons, or that writhe and twist are separately invariant. It does not claim that helicity exists for every divergence-free field, only for those whose flux form is exact, or that it is an absolute quantity without the boundary condition and the periods fixed. It does not claim that helicity is conserved in any fluid or plasma with viscosity or resistivity, or that resistive decay of helicity is always slower than that of energy; Taylor’s hypothesis is a regime, and a single force-free mode is a counterexample outside it. It does not claim a helicity conservation theorem for superfluids, or that a quantised vortex has no twist; the Seifert framing supplies one, the centreline observable sets it aside, and the sources are explicit about what has and has not been shown. It does not claim that the cross-helicity’s constancy expresses the invariance of a linking of two material line families, or that its representation as a mutual linking is wrong. It does not claim that ideal Hall MHD fails to conserve magnetic helicity; it conserves it. It does not claim that the Chern–Simons functional is a conserved charge anywhere it appears as an action, that the quantised Hall level is a linking number, that a gauge transformation can change a level, or that a fractional Hall state has a fractional internal level. It does not claim that F∧F is the second Chern form of a line bundle, whose second Chern class vanishes. It does not claim that the electromagnetic helicity is conserved by transport in general, or that no transporting flow exists; null fields have one. It does not claim that a generic light field has linked field lines. It does not claim that the α-effect is topological, that its sign is fixed by anything but the kinetic and magnetic correlations entering the chosen closure, or that the scalar α formula holds outside its closure. It does not claim that all topoisomerases change the linking number by one or two per event. It does not claim that Gauss’s note, Moreau’s note, Woltjer’s second paper, Călugăreanu’s papers, White’s paper, Chern and Simons’s paper or Hopf’s paper have been read; the reading depth of every source is stated. It does not claim that the number of times a smoke ring can be knotted has anything to do with the number of kinds of atom, which was Kelvin’s hope and is the subject of the note above. It does not claim priority for any field’s version beyond what the located sources show; the Moreau–Moffatt rediscovery is Moffatt and Tsinober’s assessment, not an inference from a missing citation.
Extensions of the structure, and what is known about each
Every entry below was searched before being written.
Next in the series
Audit 15 is defects as geometry: the statement that dislocations and disclinations in a crystal are the torsion and curvature of a connection, that the same geometry describes the defects of liquid crystals and of spacetime in Einstein–Cartan theory, and that the classification of Audit 13 and the identities of Audit 11 meet in it. It is where the lines that this page linked become the sources of a geometry, and the audit will ask in which fields the identification is a theorem and in which it is a choice of variables.
References
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Change log
0.6 · 13 Sep 2026 Erratum. Versions 0.1–0.5 stated that the fluid helicity is conserved “on a material volume or one with no boundary flux”. A material volume is not sufficient. Moffatt’s equation 17 gives dH/dt = ∮(½|u|² − h − Φ)ω·n dS for inviscid barotropic flow with conservative body force −∇Φ, so the condition is a volume bounded by a vortex surface (ω·n = 0) or, more generally, a vanishing boundary term. The counterexample, supplied in an external review of the atlas pilot built from this page and checked here: u = (−Ωy, Ωx, at), Φ = ½Ω²(x² + y²) − az, an exact Euler solution with ω = (0, 0, 2Ω), whose material volumes have helicity 2ΩatV₀. Corrected in the equation box, the ideal-fluids row, the fluid note, the dictionary, the second verdict and the HQG note; the ideal-MHD condition is stated with its fixed perfectly conducting boundary. The Moffatt reference gains the pinpoint. Published.
0.5 · 13 Sep 2026 Two corrections after a closing external review, both adopted. The plain-terms sentence “it is an integer only where something further supplies one” is replaced: the integer-valued examples are named as the Gauss linking number of two closed curves and the helicity of a suitably normalised Hopf construction, and a general real-valued helicity can equal an integer without being quantised (the two-curve integer is the linking number, and a flux-weighted helicity still needs its fluxes and self-terms specified; Q(λA) = λ²Q(A) makes the difference immediate). The Chern–Weil boundary shorthand in the abelian note and the dictionary now carries the factor through: for a global potential with F = dA on a four-manifold W, (1/4π²)∫WF∧F = (1/4π²)Q(A|∂W) by Stokes’ theorem, with the patching or extension construction retained for non-trivial bundles. Reviewer’s assessment: publication recommended on the issues audited. Published.
0.4 · 13 Sep 2026 Correction pass after a third external review, which found that several 0.3 corrections had not reached the opening, notes and dictionary; all points adopted. Opening and plain terms: an integer coupling multiplies the functional and a discrete operator spectrum does not quantise a classical value, so neither makes the functional’s value an integer; the objects are named as different. Superfluid note: Salman’s cancellation is mutual linking plus writhe against the framing’s twist (two linked planar rings); the absence of a conservation law scoped to the centreline and regularised observables, the Seifert quantity being identically zero; the first verdict and the HQG note scoped the same way. Chern–Weil integrality given with its normalisation, (1/4π²)∫F∧F = ⟨c1², [X]⟩ on a closed four-cycle, and “need not be an integer” with boundary. Hall action written in the dimensionless connection 𝒜 = eA/ħ in ledger and note, with Tong’s kT as the physical-potential coefficient; the assumptions cell and the note’s first sentence name the integer quantum Hall setting under consideration. Taylor cell: resistive rate in magnetic diffusivity ηm throughout; “bounded and not zero” removed. Dynamo: changes distinguished from values (ΔHL = −ΔHs with no flux and negligible resistive change; equal and opposite values only for near-zero initial total); “statistics of the flow” → the kinetic and magnetic correlations of the closure. Standalone legend admits the two-curve linking target; dictionary’s “nothing” and “is the response” replaced; “reasoning backwards in every row” removed. Still draft.
0.3 · 13 Sep 2026 Revised after a second external review, which found that several universal statements withdrawn in 0.2 were still standing elsewhere on the page and that four pinpoints needed correction; all adopted after checking. Replaced in place: the dek’s closed-curves sentence; “no helicity of a superfluid is a topological integer” (the mutual-linking part is discrete in units of (h/m)², the writhe is continuous, and other definitions must be checked as defined; Salman’s cancellation restated as linking-plus-writhe against the framing’s twist); “is not an integer” → “not restricted to integers”; the optical “are not” → “need not be”; “resistivity or viscosity conserves nothing” and “Resistive MHD → helicity is not conserved” → the ideal law no longer follows and the rate has no definite sign, with a zero-helicity decaying field as the counterexample; Taylor’s ordering as a hypothesis for suitable relaxing regimes; the HQG note’s “requires Kelvin’s conditions” removed; Woltjer’s theorem restated as the constrained minimum satisfying the constant-α equation, not the converse; “gauge conditions in disguise” removed; “wrong in every row” → the topology fixes the class of deformations and the dynamics decides whether the evolution stays in it. Corrections: Arnold’s asymptotic linking divides by the flow times, not the lengths (Vogel 2003, Def. 4 and Thm 9, added); Tong has not set e = ħ = 1 at eq. 5.7, his kT = e²ν/ħ carries units and the translation is given; “no spatial charge is conserved” withdrawn for the Hall row; the dynamo’s α sign is kinetic minus current helicity and the opposite-helicity balance is stated under its closed-system assumptions; the viscous rate scoped to incompressible constant-density Newtonian fluids; the electromagnetic transport failure restated as the quotient’s failure to extend, with the plane wave as the case where it does; Chern–Weil integrality placed on closed four-cycles, not manifolds with boundary; the abelian and non-abelian derivative identities given separately; Dunne’s Sec. 2.6 recognised as non-abelian and its support moved; Rudolph’s reference corrected (Lee Rudolph, Pure Appl. Math. Q. 6(2), 2010); the legend extended to a named pair of curves; the opening rewritten to give Moreau the conservation and Moffatt the interpretation; the Hall level’s integer scoped to the integer quantum Hall setting under consideration. References: 64. Still draft.
0.2 · 12 Sep 2026 Revised after external review, which found the first verdict overstated and several sources contradicted; all twelve groups adopted. The page is rebuilt on the bilinear pairing ℬ(a, b) = ∫a∧db and its quadratic form Q(a): fluid and magnetic helicity and the local abelian Chern–Simons functional are Q; cross-helicity is the mixed pairing, optical helicity two copies, DNA the two-curve case, the superfluid a filament construction, knot theory a cubic extension; “one integral” withdrawn from the dek, opening, equation box, dictionary and verdict, and the badge rule restated so that a target is named before a badge is given. Superfluids: “twist does not exist” withdrawn; Salman’s Seifert framing supplies a twist and the continuum helicity vanishes by cancellation; Kedia et al. close the relabelling route only; three observables kept apart. Chern–Simons: a gauge transformation does not change the level; on a global potential a large transformation shifts nothing; the condition on k arises for connections on non-trivial bundles, even k in general and any integer with a spin structure (Belov and Moore); “nothing is conserved” replaced by the action-versus-charge distinction; F∧F proportional to c1², not a second Chern form; the Hopf formula given with the unit-normalised area form. Quantum Hall: integer internal level for Laughlin states, response e²/2πħm, K-matrix form, Tong’s warning about integrating out the emergent field; “becomes a rational” withdrawn. MHD: Finn–Antonsen relative helicity with the mixed term; definition and conservation conditions separated; ideal Hall MHD conserves magnetic helicity (Banerjee and Galtier); resistive rate in a stated convention with a signed integrand. Cross-helicity: balance law given; mutual-linking representation retained, the material-family inference withdrawn. Electromagnetism: null fields have the transporting flow v = E×B/|B|² (Irvine 2010); magnetic helicity separately conserved for them; Hopfion “deform” sentence corrected. Relaxation: slower decay made conditional, with the single force-free mode as counterexample; “rugged” redefined; Euler–Lagrange condition distinguished from the minimiser; energy spectrum separated from the bound. Mathematics: Whitehead link caveat; mutual and self-terms separated; Arnold’s average an exact theorem; the uniqueness theorem scoped to Enciso et al. Optics: the helicity operator’s integer spectrum added to the inventory of integers. Dynamo: closure, isotropy and the diffusion and pumping terms stated. DNA: topoisomerase classes (Koster et al.); the Scientific American pagination recorded as a conflict with DOI. Moffatt’s eq. 6 relocated to Sec. 1, p. 118; Kelvin genealogy and the Moreau–Moffatt independence qualified; “oldest topological invariant” removed. References added: Banerjee and Galtier, Belov and Moore, Finn and Antonsen, Irvine 2010, Koster et al., Rudolph. Still draft.
0.1 · 12 Sep 2026 First draft on the series template. Sources verified before drafting; the verification changed the text in five places. Moffatt 1969 was read and found not to cite Moreau, and to state the two-filament helicity as αK1K2 in its abstract while its equation 6 gives 2αK1K2; both are recorded. Kedia, Kleckner, Scheeler and Irvine 2018 was found to conclude that the relabelling symmetry yields no superfluid helicity conservation law, not to define one, and the superfluid row is written accordingly. Clark di Leoni et al. compute a regularised volume helicity, not a centreline quantity, and the row says so. Steenbeck, Krause and Rädler’s abstract frames the α-effect through the Coriolis force, not through helicity, and the α–helicity formula is attributed to the later literature with its equation numbers unlocated. Bauer, Crick and White’s pagination is given as indexed. Still draft.
Method note. This page was drafted with AI assistance (Anthropic’s Claude) working from my brief and revised with me. Bibliographic details were verified against publisher records for every reference before the text was written, except for the gaps explicitly identified in the references; the flag beside each says how far the content itself was consulted, and where a claim rests on a particular passage the section, equation or page is given in the text. Claims that a topic was unexplored were searched before being kept. The verdicts are mine and I expect some of them to be argued with; corrections are welcome and will be logged. Content on this site is licensed CC BY-SA 4.0.