A higher-order analog of the helicity number for a pair of divergent-free vector fields

P. M. Akhmet'ev

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 15-23

Résumé

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.