Volume 8, issue 1 (2004)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A rational noncommutative invariant of boundary links

Stavros Garoufalidis and Andrew Kricker

Geometry & Topology 8 (2004) 115–204

arXiv: math.GT/0105028

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
Mathematical Subject Classification 2000
Primary: 57N10
Secondary: 57M25
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Publication
Received: 10 June 2002
Accepted: 16 January 2004
Published: 8 February 2004
Proposed: Robion Kirby
Seconded: Vaughan Jones, Joan Birman
Authors
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
Atlanta
Georgia 30332-0160
USA
Andrew Kricker
Department of Mathematics
University of Toronto
Toronto
Ontario
Canada M5S 3G3