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

Watanabe, Tadayuki

doi:10.2140/agt.2007.7.47

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

Tadayuki Watanabe

Algebraic & Geometric Topology 7 (2007) 47–92

DOI: 10.2140/agt.2007.7.47

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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57Q45

Secondary: 57M25

References

Publication

Received: 3 February 2006

Revised: 8 June 2006

Accepted: 12 December 2006

Published: 23 February 2007

Authors

Tadayuki Watanabe
Research Institute for Mathematical Sciences Kyoto University Kyoto Japan

Volume 7, issue 1 (2007)