Bott and Taubes considered a bundle over the space of knots whose fiber is a compactified configuration space. They produced a knot invariant by integrating along the fiber of this bundle. Their method was subsequently used to construct all finite-type (Vassiliev) knot invariants as well as real cohomology classes in spaces of knots in R^n, n > 3. Replacing integration of differential forms by a Pontrjagin-Thom construction, I have produced cohomology classes with arbitrary coefficients. Motivated by work of Budney and F. Cohen on the homology of the space of long knots in R^3, I have also proven a product formula for these classes with respect to connect-sum. If time permits, I will mention some results towards understanding these classes using tools coming from the Goodwillie-Weiss embedding calculus, as well as some further refinements in progress.