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Zapiski Nauchnykh Seminarov POMI, 2001, Volume 279, Pages 15–23
(Mi znsl1451)
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This article is cited in 2 scientific papers (total in 2 papers)
A higher-order analog of the helicity number for a pair of divergent-free vector fields
P. M. Akhmet'ev Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation
Abstract:
Pairs $B$, $\tilde B$ of divergent-free vector fields with compact support in $\mathbb R^3$ are considered.
A higher-order analog $M(B,\tilde B)$ (of order 3) of the Gauss helicity number $H(B,\tilde B)=\int A\tilde
B\,d\mathbb R^3$, $\operatorname{curl}(A)=B$, (of order 1) is constructed, which is invariant under
volume-preserving diffeomorphisms. An integral expression for $M$ is given. A degree-four polynomial $m(B(x_1)$, $B(x_2)$, $\tilde B(\tilde x_1)$, $\tilde B(\tilde x_2))$, $x_1$, $x_2$, $\tilde x_1$, $\tilde x_2\in\mathbb R^3$, is defined, which is symmetric in the first and second pairs of variables separately.
$M$ is the average value of $m$ over arbitrary configurations of points. Several conjectures clarifying
the geometric meaning of the invariant and relating it with invariants of knots and links are stated.
Received: 25.01.2001
Citation:
P. M. Akhmet'ev, “A higher-order analog of the helicity number for a pair of divergent-free vector fields”, Geometry and topology. Part 6, Zap. Nauchn. Sem. POMI, 279, POMI, St. Petersburg, 2001, 15–23; J. Math. Sci. (N. Y.), 119:1 (2004), 5–9
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https://www.mathnet.ru/eng/znsl1451 https://www.mathnet.ru/eng/znsl/v279/p15
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