Generalized Bott–Cattaneo–Rossi invariants in terms of Alexander polynomials

Abstract

The Bott–Cattaneo–Rossi invariant $(Z_k)_{k \in \mathbb{N} \setminus \{0, 1\}}$ is an invariant of long knots $\mathbb{R}^{n} \hookrightarrow \mathbb{R}^{n+2}$ for odd $n$, which reads as a combination of integrals over configuration spaces. In this article, we compute such integrals and prove explicit formulas for (generalized) $Z_{k}$ in terms of Alexander polynomials, or in terms of linking numbers of some cycles of a hypersurface bounded by the knot. Our formulas, which hold for all null-homologous long knots in homology $\mathbb{R}^{n+2}$ at least when $n \equiv 1 \mod 4$, conversely express the Reidemeister torsion of the knot complement in terms of $(Z_{k})_{k \in \mathbb{N} \setminus \{0, 1\}}$. Our formula extends to the even-dimensional case, where $Z_{k}$ will be proved to be well-defined in an upcoming article.