  This work continues the study of a homotopy-theoretic construction of the
author inspired by the Bott-Taubes integrals. Bott and Taubes constructed knot
invariants by integrating differential forms along the fiber of a bundle over
the space of knots. Their techniques were later used by Cattaneo et al. to
construct real "Vassiliev-type" cohomology classes in spaces of knots in
higher-dimensional Euclidean space. By doing this integration via a
Pontrjagin-Thom construction, we constructed cohomology classes in the knot
space with arbitrary coefficients. We later showed that a refinement of this
construction recovers the Milnor triple linking number for string links. We
conjecture that we can produce all Vassiliev-type classes in this manner. Here
we extend our homotopy-theoretic constructions to the stages of the Taylor
tower for the knot space, which arises from the Goodwillie-Weiss embedding
calculus. We use the model of "punctured knots and links" for the Taylor tower.
