Generalized Gauss Maps and Integrals for Three-Component Links: Toward Higher Helicities for Magnetic Fields and Fluid Flows

To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientation-preserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space

Generalized Gauss Maps and Integrals for Three-Component Links: Toward Higher Helicities for Magnetic Fields and Fluid Flows

To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientation-preserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space

Generalized Gauss Maps and Integrals for Three-Component Links: Toward Higher Helicities for Magnetic Fields and Fluid Flows

To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientation-preserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space

Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows, Part 2

We describe a new approach to triple linking invariants and integrals, aiming for a simpler, wider and more natural applicability to the search for higher order helicities of fluid flows and magnetic fields. To each three-component link in Euclidean 3-space, we associate a geometrically natural generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers, but patterned after J.H.C. Whitehead's integral formula for the Hopf invariant. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of 3-space, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus. In the first paper of this series [math.GT 1101.3374] we did this for three-component links in the 3-sphere. Komendarczyk has applied this approach in special cases to derive a higher order helicity for magnetic fields whose ordinary helicity is zero, and to obtain from this nonzero lower bounds for the field energy.Comment: 22 pages, 8 figures. arXiv admin note: text overlap with arXiv:1101.337

Pontryagin invariants and integral formulas for Milnor's triple linking number

To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus.Comment: 60 pages, 37 figure

International Consensus Statement on Rhinology and Allergy: Rhinosinusitis

Background: The 5 years since the publication of the first International Consensus Statement on Allergy and Rhinology: Rhinosinusitis (ICAR‐RS) has witnessed foundational progress in our understanding and treatment of rhinologic disease. These advances are reflected within the more than 40 new topics covered within the ICAR‐RS‐2021 as well as updates to the original 140 topics. This executive summary consolidates the evidence‐based findings of the document. Methods: ICAR‐RS presents over 180 topics in the forms of evidence‐based reviews with recommendations (EBRRs), evidence‐based reviews, and literature reviews. The highest grade structured recommendations of the EBRR sections are summarized in this executive summary. Results: ICAR‐RS‐2021 covers 22 topics regarding the medical management of RS, which are grade A/B and are presented in the executive summary. Additionally, 4 topics regarding the surgical management of RS are grade A/B and are presented in the executive summary. Finally, a comprehensive evidence‐based management algorithm is provided. Conclusion: This ICAR‐RS‐2021 executive summary provides a compilation of the evidence‐based recommendations for medical and surgical treatment of the most common forms of RS

Vela-Vick, Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links

Abstract. Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer µ, the "triple linking number" of the title, which is well-defined modulo the greatest common divisor of p, q and r. The Borromean rings: p = q = r = 0, µ = ±1 To each such link L we associate a geometrically natural characteristic map gL from the 3-torus to the 2-sphere in such a way that link homotopies of L become homotopies of gL. Maps of the 3-torus to the 2-sphere were classified up to homotopy by Lev Pontryagin in 1941. A complete set of invariants is given by the degrees p, q and r of their restrictions to the 2-dimensional coordinate subtori, and by the residue class of one further integer ν, an "ambiguous Hopf invariant" which is well-defined modulo twice the greatest common divisor of p, q and r. We show that the pairwise linking numbers p, q and r of the components of L are equal to the degrees of its characteristic map gL restricted to the 2-dimensional subtori, and that twice Milnor's µ-invariant for L is equal to Pontryagin's ν-invariant for gL. When p, q and r are all zero, the µ-and ν-invariants are ordinary integers. In this case we use J. H. C. Whitehead's integral formula for the Hopf invariant, adapted to maps of the 3-torus to the 2-sphere, together with a formula for the fundamental solution of the scalar Laplacian on the 3-torus as a Fourier series in three variables, to provide an explicit integral formula for ν, and hence for µ. We give here only sketches of the proofs of the main results, with full details to appear elsewhere. Statement of results Consider the configuration space of ordered triples (x, y, z) of distinct points in the unit 3-sphere S 3 in R 4 . Since x, y and z are distinct, they span a 2-plane in R 4 . Orient this plane so that the vectors from x to y and from x to z form a positive basis, and then move it parallel to itself until it passes through the origin. The result is an element G(x, y, z) of the Grassmann manifold G 2 R 4 of all oriented 2-planes through the origin in R 4 . This defines the Grassmann map It is equivariant with respect to the diagonal O(4) action on S 3 × S 3 × S 3 and the usual O(4) action on G 2 R 4 . G(x, y, z) The Grassmann manifold G 2 R 4 is isometric (up to scale) to the product S 2 × S 2 of two unit 2-spheres. Let π : G 2 R 4 → S 2 denote orthogonal projection to either factor. Given any ordered oriented link L in S 3 with three parametrized components ). In Section 3 we give an explicit formula for this map as the unit normalization of a vector field on T 3 whose components are quadratic polynomials in the components of x(s), y(t) and z(u). The homotopy class of g L is unchanged under any link homotopy of L, meaning a deformation during which each component may cross itself, but different components may not intersect. Triple linking numbers, ambiguous Hopf invariants and integral formulas Theorem A. The pairwise linking numbers p, q and r of the link L are equal to the degrees of its characteristic map g L on the 2-dimensional coordinate subtori of T 3 , while twice Milnor's µ-invariant for L is equal to Pontryagin's ν-invariant for g L . Remark. Milnor's µ-invariant, typically denoted µ 123 , is descriptive of a single threecomponent link. In contrast, Pontryagin's ν-invariant is the cohomology class of a difference cocycle comparing two maps from T 3 to S 2 that are homotopic on the 2-skeleton of T 3 . In particular, it assigns to any pair g, g of such maps whose degrees on the coordinate 2-tori are p, q and r, an integer ν(g, g ) that is well-defined modulo 2 gcd(p, q, r). With this understanding, the last statement in Theorem A asserts that for any two links L and L with the same pairwise linking numbers p, q and r. We will sketch here two quite different proofs of Theorem A, a topological one in Section 4 using framed cobordism of framed links in the 3-torus, and an algebraic one in Section 5 using the group of link homotopy classes of three-component string links and the fundamental groups of spaces of maps of the 2-torus to the 2-sphere. To state the integral formula for Milnor's µ-invariant when the pairwise linking numbers are zero, let ω denote the Euclidean area 2-form on S 2 , normalized so that the total area is 1 instead of 4π. Then ω pulls back under the characteristic map g L to a closed 2-form on T 3 , which can be converted in the usual way to a divergence-free vector field V L on T 3 . In Section 6 we give explicit formulas for V L , and also for the fundamental solution ϕ of the scalar Laplacian on the 3-torus as a Fourier series in three variables. These are the key ingredients in the integral formula below. Theorem B. If the pairwise linking numbers p, q and r of the three components of L are all zero, then Milnor's µ-invariant of L is given by the formula Here ∇ σ indicates the gradient with respect to σ, the difference σ − τ is taken in the abelian group structure of the torus, and dσ and dτ are volume elements. The integrand is invariant under the action of the group SO(4) of orientation-preserving rigid motions of S 3 on the link L, attesting to the naturality of the formula. We will see in the next section that the integral above expresses the "helicity" of the vector field V L on T 3 . Background and motivation Let L be an ordered oriented link in R 3 with two parametrized components The classical linking number Lk(X, Y ) is the degree of the Gauss map S 1 × S 1 → S 2 sending (s, t) to (y(t)−x(s))/ y(t)−x(s) , and can be expressed by the famous integral formula of Gauss [1833], where ϕ(r) = −1/(4πr) is the fundamental solution of the scalar Laplacian in R 3 . The integrand is invariant under the group of orientation-preserving rigid motions of R 3 , acting on the link L. Corresponding formulas in S 3 appear in DeTurck and Gluck [2008] and in Kuperberg [2008]. Theorems A and B above give a similar formulation of Milnor's triple linking number in S 3 . We emphasize that these two theorems are set specifically in S 3 , and that so far we have been unable to find corresponding formulas in Euclidean space R 3 which are equivariant (for Theorem A) and invariant (for Theorem B) under the noncompact group of orientation-preserving rigid motions of R 3 . For some background on higher order linking invariants, see Milnor [1957] and, for example, Massey The helicity of a vector field V defined on a bounded domain Ω in R 3 is given by the formula where, as above, ϕ is the fundamental solution of the scalar Laplacian on R 3 . Woltjer [1958] introduced this notion during his study of the magnetic field in the Crab Nebula, and showed that the helicity of a magnetic field remains constant as the field evolves according to the equations of ideal magnetohydrodynamics, and that it provides a lower bound for the field energy during such evolution. The term "helicity" was coined by Moffatt [1969], who also derived the above formula. There is no mistaking the analogy with Gauss's linking integral, and no surprise that helicity is a measure of the extent to which the orbits of V wrap and coil around one another. Since its introduction, helicity has played an important role in astrophysics and solar physics, and in plasma physics here on earth. 4 Triple linking numbers, ambiguous Hopf invariants and integral formulas Looking back at Theorem B, we see that the integral in our formula for Milnor's µ-invariant of a three-component link L in the 3-sphere expresses the helicity of the associated vector field V L on the 3-torus. Our study was motivated by a problem proposed by Arnol d and Khesin [1998] regarding the search for "higher helicities" for divergence-free vector fields. In their own words: The dream is to define such a hierarchy of invariants for generic vector fields such that, whereas all the invariants of order ≤ k have zero value for a given field and there exists a nonzero invariant of order k + 1, this nonzero invariant provides a lower bound for the field energy. Many others have been motivated by this problem, and have contributed to its understanding; see, for example