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

Abstract

The Bott–Cattaneo–Rossi invariant (𝑍_𝑘)_{𝑘 ∈ ℕ ⧵ {0, 1}} is an invariant of long knots ℝ^𝑛 ↪ 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 ℝ^𝑛+2 at least when 𝑛 ≡ 1 mod 4, conversely express the Reidemeister torsion of the knot complement in terms of (𝑍_𝑘)_{𝑘 ∈ ℕ ⧵ {0, 1}}. Our formula extends to the even-dimensional case, where 𝑍_𝑘 will be proved to be well-defined in an upcoming article.