Configuration space integral for long $n$–knots and the Alexander polynomial

Abstract

There is a higher dimensional analogue of the perturbative Chern–Simons theory in the sense that a similar perturbative series as in 3 dimensions, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott–Cattaneo–Rossi invariant). This invariant was constructed by Bott for degree 2 and by Cattaneo–Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon $n$–knots and characterize the Bott–Cattaneo–Rossi invariant as a finite type invariant of long ribbon $n$–knots introduced by Habiro–Kanenobu–Shima. As a consequence, we obtain a nontrivial description of the Bott–Cattaneo–Rossi invariant in terms of the Alexander polynomial.