Given a link $L$ in $R^3$ with just two components $X = \{x(s):s \epsilon S^1 \}$ and $Y = \{y(t): t\epsilon S^1\}$, their linking number can be defined as the degree of the Gauss map $f_L: S^1 \times S^1\rightarrow S^2$ , given by $f_L(s,t)=(y(t)-x(s))/|y(t)-x(s)|$, or as the value of the Gauss integral, $1/4\pi\int_{S1\times S1} (dx/ds) x (dy/dt)$ • $(x y)/|x y|^3 ds dt$.The Gauss map is equivariant with respect to the action of an orientation-preserving isometry $h$ of $R^3$ , that is, $f_h(L) = h\circ f_L$, and the Gauss integrand is invariant with respect to this action, meaning that it is the same for $L$ and for $h(L)$ , attesting to the geometric naturality of both the map and the integral.
A consequence of this naturality is that the Gauss integral hides inside Ampere's Law of classical electrodynamics, and appears explicitly in the notions of helicity in plasma physics and fluid dynamics, and of the writhing number of a simple closed curve, used in molecular biology.
We quickly review the above ideas, constructions and applications, and then turn to the corresponding notions for three-component links $L$ in Euclidean 3-space $R^3$ and the 3-sphere $S^3$ .
In both cases we introduce generalizations $f_L:S^1\times S^1\times S^1 \rightarrow S^2$ of the Gauss map which share its essential features: simplicity, naturality in the above sense, and this time a transfer of the classical Milnor invariants for three-component links up to link homotopy to the classical Pontryagin invariants for the characteristic map $f_L$ up to homotopy. Then, when the pairwise linking numbers are all zero, we use this map to derive an integral formula for Milnor's triple linking number which is a natural successor to the Gauss integral above. Core proofs are in terms of framed cobordismin three-manifolds.
Motivation comes from the Arnold-Khesin search for "higher order helicities" of plasma and fluid flows.
All of this is joint work with Dennis DeTurck, Rafal Komendarczyk, Paul Melvin, Haggai Nuchi, Clayton Shonkwiler and Shea Vela-Vick.