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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.
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