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.
Notes on file formats
Stavros Garoufalidis, Andrew Kricker
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
and
Department of Mathematics, University of Toronto
Toronto, Ontario, Canada M5S 3G3
Email: stavros@math.gatech.edu, akricker@math.toronto.edu
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