Volume 4, issue 2 (2004)

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A computation of the Kontsevich integral of torus knots

Julien Marche

Algebraic & Geometric Topology 4 (2004) 1155–1175

arXiv: math.GT/0404264

Abstract

We study the rational Kontsevich integral of torus knots. We construct explicitely a series of diagrams made of circles joined together in a tree-like fashion and colored by some special rational functions. We show that this series codes exactly the unwheeled rational Kontsevich integral of torus knots, and that it behaves very simply under branched coverings. Our proof is combinatorial. It uses the results of Wheels and Wheeling and various spaces of diagrams.

Keywords
finite type invariants, Kontsevich integral, torus knots, Wheels, Wheeling, rationality
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57M25, 57R56
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Publication
Received: 6 May 2004
Revised: 8 November 2004
Accepted: 15 November 2004
Published: 10 December 2004
Authors
Julien Marche
Institut de Mathématiques de Jussieu
Équipe “Topologie et Géométries Algébriques”
Case 7012
Université Paris VII
75251 Paris Cedex 05
France