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Rationality: From Lie algebras to Lie groups

On the level of Lie algebras, the Kontsevich integral of a knot (a graph-valued invariant) becomes the colored Jones function (a power series invariant). Rozansky conjectured and the authors proved a Rationality Conjecture for the Kontsevich integral. In this note, we explain how the Rationality Conjecture is related to Lie groups.