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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.

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 58J28 Eta-invariants, Chern-Simons invariants
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