Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows, part II

Abstract

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.

To each three-component link in Euclidean $3$–space, 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. This generalized Gauss map is a natural successor to Gauss’s original map from the $2$–torus to the $2$–sphere. Like its prototype, it is equivariant with respect to orientation-preserving isometries of the ambient space, attesting to its naturality and positioning it for application to physical situations.

When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number which is a natural successor to the classical Gauss integral for the pairwise linking numbers, with an integrand invariant under orientation-preserving isometries of the ambient space. This new integral is patterned after J H C Whitehead’s integral formula for the Hopf invariant, and hence interpretable as the ordinary helicity of a related vector field on the $3$–torus.