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

TL;DR

This paper extends homotopy-theoretic constructions inspired by Bott-Taubes integrals to the Taylor tower of knot spaces, aiming to produce all Vassiliev-type classes through a refined Pontrjagin-Thom approach.

Contribution

It develops a homotopy-theoretic framework for the Taylor tower of knot spaces, connecting Bott-Taubes integrals with Goodwillie-Weiss calculus.

Findings

Constructs cohomology classes in the Taylor tower stages.

Recovers Milnor triple linking number for string links.

Proposes a conjecture to produce all Vassiliev classes.

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. 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.