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

## TL;DR

This paper generalizes the Bott-Cattaneo-Rossi invariants for high-dimensional long knots, providing a flexible diagram-counting framework that extends to knots in broader manifolds.

## Contribution

It introduces a new, adaptable definition of these invariants, enabling their interpretation as diagram counts and extending their applicability to more general manifolds.

## Key findings

- Invariants can be interpreted as counts of diagrams.
- Extension of invariants to knots in asymptotic homology manifolds.
- Provides a flexible framework for high-dimensional knot invariants.

## 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 $\geq 3$. 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.