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

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 $M = ℂ^{*}$ , it leads for each positive integer $N$ to a filtration on the space of tangles in $ℂ^{*} \times I$ (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.

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Keywords

surface knots, finite type invariants, quantum invariants, knot theory, Vassiliev invariants, quantum algebra, links in solid torus

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 17B37

References

Publication

Received: 18 September 2012

Revised: 17 April 2013

Accepted: 23 April 2013

Published: 10 October 2013

Authors

Adrien Brochier
Université de Genève Bd du Pont-d’Arve 40 CH-1211 Geneve Switzerland
http://abrochier.org

Volume 13, issue 6 (2013)

Adrien Brochier

Abstract