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
–knots and
characterize the Bott–Cattaneo–Rossi invariant as a finite type invariant of long ribbon
–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.
Keywords
configuration space integral, ribbon $n$–knots, Alexander
polynomial, finite type invariant
