In a previous article, we gave a more flexible definition of an invariant $(Z_k)_{k\in \mathbb N\setminus\{0,1\}}$ of Bott, Cattaneo, and Rossi, which is a combination of integrals over configuration spaces for long knots $\mathbb R^n\hookrightarrow\mathbb R^{n+2}$, for odd $n\geq 3$. This extended the definition of the invariant $(Z_k)_{k\in\mathbb N\setminus\{0,1\}}$ to all long knots in asymptotic homology $\mathbb R^{n+2}$, for odd $n\geq3$. In this article, we obtain a formula for $Z_k$ in terms of linking numbers of some cycles of a surface bounded by the knot and we express the Reidemeister torsion of the knot complement in terms of $(Z_k)_{k\in\mathbb N\setminus\{0,1\}}$, when $n\equiv1\mod 4$.