Relating a Bott-Cattaneo-Rossi invariant to Alexander polynomials

David Leturcq

Published 2020-03-02Version 1

In a previous article, we gave a more flexible definition of an invariant $(Z_k)_{k\in \mathbb N\setminus\{0\}}$ of Bott, Cattaneo, and Rossi defined using integration on 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 $(Z_k)_{k\in\mathbb N\setminus\{0\}}$ to all long knots in asymptotic homology $\mathbb R^{n+2}$, for odd $n\geq3$. In this article, we obtain a formula for $Z_2$ in terms of linking numbers of some cycles of a surface bounded by the knot, when $n\equiv \mod 4$. This yields a relation between $Z_2$ and Alexander polynomials.