The Kontsevich integral for bottom tangles in handlebodies
Résumé
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: B -> (A) over cap, where B is the category of bottom tangles in handlebodies and (A) over cap is the degree-completion of the category A of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, 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 gr B congruent to A, where gr B is the associated graded of the Vassiliev-Goussarov filtration on B. To each Drinfeld associator phi we associate a ribbon quasi-Hopf algebra H-phi , in (A) over cap, 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 B. Finally, we explain how Z refines the LMO functor, which is a TQFT-like functor extending the Le-Murakami-Ohtsuki invariant.