On the functors associated with beaded open Jacobi diagrams

TL;DR

This paper explores the functors derived from Jacobi diagrams in handlebodies, analyzing their polynomiality and outer properties, and generalizes earlier findings by Katada within a categorical framework.

Contribution

It establishes new results on the polynomiality and outer nature of functors associated with beaded open Jacobi diagrams, extending previous work by Katada.

Findings

Demonstrates polynomiality of the functors

Shows conditions under which functors are outer

Generalizes Katada's results on Jacobi diagram functors

Abstract

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