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Morphisms in the linear category A of Jacobi diagrams in handlebodies giverise to interesting contravariant functors on the category gr offinitely-generated free groups, encoding part of the composition structure ofthe category A. These functors correspond, via an equivalence of categoriesgiven by Powell, to functors given by beaded open Jacobi diagrams. We study thepolynomiality of these functors and whether they are outer functors. Theseresults are inspired by and generalize previous results obtained by Katada.

On the functors associated with beaded open Jacobi diagrams