Cyclotomic associators and finite type invariants for tangles in the solid torus

Abstract

The universal Vassiliev–Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded functor is an isomorphism. The Vassiliev filtration has a natural extension to tangles in any thickened surface $M\times I$ but the corresponding category of diagrams lacks some finiteness properties which are essential to the above construction. We suggest to overcome this obstruction by studying families of Vassiliev invariants which, roughly, are associated to finite coverings of $M$. In the case $M={ℂ}^{\ast }$, it leads for each positive integer $N$ to a filtration on the space of tangles in ${ℂ}^{\ast }\times I$ (or “$B$–tangles”). We first prove an extension of the Shum–Reshetikhin–Turaev Theorem in the framework of braided module category leading to $B$–tangles invariants. We introduce a category of “$N$–chord diagrams”, and use a cyclotomic generalization of Drinfeld associators, introduced by Enriquez, to put a braided module category structure on it. We show that the corresponding functor from the category of $B$–tangles is a universal invariant with respect to the $N$ filtration. We show that Vassiliev invariants in the usual sense are well approximated by $N$ finite type invariants. We show that specializations of the universal invariant can be constructed from modules over a metrizable Lie algebra equipped with a finite order automorphism preserving the metric. In the case the latter is a “Cartan” automorphism, we use a previous work of the author to compute these invariants explicitly using quantum groups. Restricted to links, this construction provides polynomial invariants.