In our paper “Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows, part II”, Dennis DeTurck, Herman Gluck, Rafal Komendarczyk, Paul Melvin, Haggai Nuchi, David Shea Vela-Vick and myself defined a generalization of the Gauss map which detects all link-homotopy invariants of three-component links in \(\mathbb{R}^3\).
The usual Gauss map is defined as follows: if \(X = \{x(s): s \in S^1\}\) and \(Y=\{y(t):t \in S^1\}\) form a two-component link, then the Gauss map is the map \(g: S^1 \times S^1 \to S^2\) given by \(g(s,t) = \frac{x(s)-y(t)}{\|x(s)-y(t)\|}\). The degree of the Gauss map is equal to the linking number of the link, and computing that degree explicitly yields the famous Gauss linking integral:
\(Lk(X,Y) = \frac{1}{4\pi} \int_{S^1 \times S^1} \frac{dx}{ds} \times \frac{dy}{dt} \cdot \frac{x(s) - y(t)}{\|x(s)-y(t)\|^3} ds dt\)
In turn, this integral is the key to many physical applications, especially the helicity of a plasma or fluid flow.
Our generalized Gauss map applies to three-component links. Given parametrized components \(X = \{x(s): s \in S^1\}\), \(Y=\{y(t):t \in S^1\}\), and \(Z = \{z(u): u \in S^1\}\), a triple of points \(x(s), y(t),z(u)\), one from each component, forms a (possibly degenerate) triangle, as in the image above. Letting \(n\) be the unit normal to this triangle, we define the generalized Gauss map \(G: S^1 \times S^1 \times S^1 \to S^2\) to be the normalization of
\(\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + (\sin \alpha + \sin \beta + \sin \gamma)n\)
We show that the homotopy invariants of this map are precisely the link-homotopy invariants of the three-component link. In particular, the degrees of its restrictions to the faces of the torus recover the pairwise linking numbers (so it really is a generalization of the Gauss map), and its Hopf invariant gives Milnor’s triple linking number of the link. In turn, this yields a Gauss-type integral formula for the triple linking number.