Geometry & Topology, Vol. 8 (2004) Paper no. 4, pages 115--204.

A rational noncommutative invariant of boundary links

Stavros Garoufalidis and Andrew Kricker

Abstract. In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky's conjecture was soon proven by the second author. We begin our paper by reviewing Rozansky's conjecture and the main ideas that lead to its proof. The natural question of extending this conjecture to links leads to the class of boundary links, and a proof of Rozansky's conjecture in this case. A subtle issue is the fact that a `hair' map which replaces beads by the exponential of hair is not 1-1. This raises the question of whether a rational invariant of boundary links exists in an appropriate space of trivalent graphs whose edges are decorated by rational functions in noncommuting variables. A main result of the paper is to construct such an invariant, using the so-called surgery view of boundary links and after developing a formal diagrammatic Gaussian integration.
Since our invariant is one of many rational forms of the Kontsevich integral, one may ask if our invariant is in some sense canonical. We prove that this is indeed the case, by axiomatically characterizing our invariant as a universal finite type invariant of boundary links with respect to the null move. Finally, we discuss relations between our rational invariant and homology surgery, and give some applications to low dimensional topology.

Keywords. Boundary links, Kontsevich integral, Cohn localization

AMS subject classification. Primary: 57N10. Secondary: 57M25.

DOI: 10.2140/gt.2004.8.115

E-print: arXiv:math.GT/0105028

Submitted to GT on 10 June 2002. Paper accepted 16 January 2004. Paper published 8 February 2004.

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Stavros Garoufalidis, Andrew Kricker
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
Department of Mathematics, University of Toronto
Toronto, Ontario, Canada M5S 3G3


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