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

Watanabe, Tadayuki¹

¹Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 47-92

Résumé

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.

Détail

DOI : 10.2140/agt.2007.7.47

Keywords: configuration space integral, ribbon $n$–knots, Alexander polynomial, finite type invariant

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