The Kontsevich integral for bottom tangles in handlebodies

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 $Z:\mathcal{B}\to \hat{A}$, where $\mathcal{B}$ is the category of bottom tangles in handlebodies and $\hat{A}$ is the degree-completion of the category $\mathbf{A}$ of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, $\mathbf{A}$ 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 $\mathrm{gr}\mathcal{B}\cong \mathbf{A}$, where $\mathrm{gr}\mathcal{B}$ is the associated graded of the Vassiliev–Goussarov filtration on $\mathcal{B}$. To each Drinfeld associator $\phi$ we associate a ribbon quasi-Hopf algebra $H_{\phi }$ in $\hat{A}$, and we prove that the braided Hopf algebra resulting from $H_{\phi }$ 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.