A Higher-Order Analog of the Linking Number for a Pair of Divergence-Free Vector Fields

Abstract

Pairs B, \(\tilde B\) of divergence-free vector fields with compact support in \(\mathbb{R}^3 \) are considered higher-order analog M(B,\(\tilde B\)c (of order 3) of the Gauss helicity number H(B, \(\tilde B\))=\(\smallint A\widetilde{\rm B}d\mathbb{R}^3 \), 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(x1), B(x2), \(\tilde B\)(\(\tilde x\) 1), \(\tilde B\)(\(\tilde x\) 2)), x1, x2, \(\tilde x\) 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 to invariants of knots and links are stated. Bibliography: 11 titles.