  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 of fluid flows and magnetic fields. To each three-component
link in Euclidean 3-space, we associate a geometrically natural 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 can be viewed as a natural extension of the
familiar fact that the linking number of a two-component link in 3-space is the
degree of its associated Gauss map from the 2-torus to the 2-sphere. When the
pairwise linking numbers are all zero, we give an integral formula for the
triple linking number analogous to the Gauss integral for the pairwise linking
numbers, but patterned after J.H.C. Whitehead's integral formula for the Hopf
invariant. The integrand in this formula is geometrically natural in the sense
that it is invariant under orientation-preserving rigid motions of 3-space,
while the integral itself can be viewed as the helicity of a related vector
field on the 3-torus. In the first paper of this series [math.GT 1101.3374] we
did this for three-component links in the 3-sphere. Komendarczyk has applied
this approach in special cases to derive a higher order helicity for magnetic
fields whose ordinary helicity is zero, and to obtain from this nonzero lower
bounds for the field energy.
