Homotopy Bott–Taubes integrals and the Taylor tower for spaces of knots and links

Abstract

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 in $$\mathbb {R}^3$$ . Their techniques were later used by Cattaneo et al. to construct real “Vassiliev-type” cohomology classes in the space of knots in $$\mathbb {R}^d$$ , $$d\ge 4$$ . 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 embedding space, which arises from the Goodwillie–Weiss embedding calculus. We use the model of “punctured knots and links” for the Taylor tower.