Bott and Taubes constructed knot invariants by integrating differential forms along
the fiber of a bundle over the space of knots, generalizing the Gauss linking integral.
Their techniques were later used to construct real cohomology classes in spaces of
knots and links in higher-dimensional Euclidean spaces. In previous work, we
constructed cohomology classes in knot spaces with arbitrary coefficients by
integrating via a Pontrjagin–Thom construction. We carry out a similar construction
over the space of string links, but with a refinement in which configuration spaces are
glued together according to the combinatorics of weight systems. This gluing is
somewhat similar to work of Kuperberg and Thurston. We use a formula of Mellor
for weight systems of Milnor invariants, and we thus recover the Milnor
triple linking number for string links, which is in some sense the simplest
interesting example of a class obtained by this gluing refinement of our previous
methods. Along the way, we find a description of this triple linking number as a
degree of a map from the 6–sphere to a quotient of the product of three
2–spheres.
Keywords
triple linking number, configuration space integrals,
Pontrjagin–Thom construction, gluing, degree, string links,
finite-type link invariants
