On the holomorphic point of view in the theory of quantum knot invariants

In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from a geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of flat SU(2)-connections on the torus. Two results are emphasized: the reconstruction from Weyl quantization of the restriction to the torus of the modular functor, and a description of a basis of the space of quantum observables on the torus in terms of colored curves, which answers a question related to quantum computing. (c) 2005 Elsevier B.V. All rights reserved.