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
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
. In the case
, it leads for each positive
integer to a filtration on
the space of tangles in
(or “–tangles”).
We first prove an extension of the Shum–Reshetikhin–Turaev
Theorem in the framework of braided module category leading to
–tangles invariants. We
introduce a category of “–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
–tangles is a universal
invariant with respect to the
filtration. We show that Vassiliev invariants in the usual sense are well approximated
by
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.
Keywords
surface knots, finite type invariants, quantum invariants,
knot theory, Vassiliev invariants, quantum algebra, links
in solid torus
