# The Kontsevich integral for bottom tangles in handlebodies

## TL;DR

This paper extends the Kontsevich integral to bottom tangles in handlebodies, constructing a functor that links topological objects with algebraic structures, and refines the LMO invariant within a TQFT framework.

## Contribution

It introduces a functor from bottom tangles in handlebodies to Jacobi diagrams, establishing an isomorphism with the associated graded of the Vassiliev-Goussarov filtration, and connects Drinfeld associators to ribbon quasi-Hopf algebras.

## Key findings

- Constructed a functor Z:  o \u00a0 for bottom tangles in handlebodies.
- Proved an isomorphism  ext{ gr}  e7 .
- Linked Drinfeld associators to ribbon quasi-Hopf algebras in the graded setting.

## Abstract

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z:\mathcal{B}\to \widehat{\mathbb{A}}$, where $\mathcal{B}$ is the category of bottom tangles in handlebodies and $\widehat{\mathbb{A}}$ is the degree-completion of the category $\mathbb{A}$ of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, $\mathbb{A}$ is the linear PROP governing "Casimir Hopf algebras", which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2-tensor. The functor $Z$ induces a canonical isomorphism $\hbox{gr}\mathcal{B} \cong \mathbb{A}$, where $\hbox{gr}\mathcal{B}$ is the associated graded of the Vassiliev-Goussarov filtration on $\mathcal{B}$. To each Drinfeld associator $\varphi$ we associate a ribbon quasi-Hopf algebra $H_\varphi$ in $\hbox{gr}\mathcal{B}$, and we prove that the braided Hopf algebra resulting from $H_\varphi$ by "transmutation" is precisely the image by $Z$ of a canonical Hopf algebra in the braided category $\mathcal{B}$. Finally, we explain how $Z$ refines the LMO functor, which is a TQFT-like functor extending the Le-Murakami-Ohtsuki invariant.