Generalized Bott-Cattaneo-Rossi invariants of high-dimensional long knots

Abstract

Bott, Cattaneo and Rossi defined invariants of long knots ${\mathbb{R}}^n\hookrightarrow {\mathbb{R}}^{n+2}$ as combinations of configuration space integrals for $n$ odd $\ge 3$. Here, we give a more flexible definition of these invariants. Our definition allows us to interpret these invariants as counts of diagrams. It extends to long knots inside more general $(n+2)$-manifolds, called asymptotic homology ${\mathbb{R}}^{n+2}$, and provides invariants of these knots.

47 pages; updated after referee process; extension of BCR invariants to non-parallelizable spaces; simplest version of the proof of the additivity under connected sum ; the bibliography and the introduction have been updated with references to an upcoming version of arXiv:2003.01007, and an upcoming article about the 1-dimensional case