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About the uniqueness of the Kontsevich integral. (English) Zbl 1027.57016
The paper under review proves a certain uniqueness theorem of the Kontsevich integral, which generalizes the one by T. T. Q. Le and J. Murakami [Compos. Math. 102, 41-64 (1996; Zbl 0851.57007)]. Recall that the Kontsevich integral can be defined as a monoidal functor \(Z^K: T\to A\) from the category \(T\) of \(q\)-tangles to a category \(A\) of diagrams. The author proves that if a monoidal functor \(Z: T\to A\) satisfies a certain set of natural conditions, then \(Z\) recovers from the Kontsevich integral \(Z^K\) and an element \(a^Z\) of \(A\) involving only two-leg diagrams, i.e., diagrams with only two univalent vertices. The Kontsevich integral \(Z^K\) and the limit \(K^l\) of the configuration space integral defined by S. Poirier [Algebr. Geom. Topol. 2, 1001-1050 (2002; Zbl 1024.57015)] are both monoidal functors satisfying the conditions required in the main theorem, and it follows as a corollary that the anomaly of D. Altschuler and L. Freidel [Commun. Math. Phys. 187, 261-287 (1997; Zbl 0949.57012)] involves only two-leg diagrams.

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
58J28 Eta-invariants, Chern-Simons invariants
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