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.