| The Reshetikhin-Turaev functor, when restricted to D , the subcategory of tangled ribbon graphs consisting of links and tangled trivalent graphs colored by tensor powers of the fundamental representation of Uq( g2 ), determines a polynomial valued, regular isotopy invariant of such graphs. The Kuperberg-Jaeger skein relations provide an inductive, combinatorial method of calculating this invariant. We show that in the quotient category of D with respect to these skein relations, the identity morphisms corresponding to 1, 2 and 3 parallel strands are equivalent to a sum of idempotent, orthogonal morphisms called projectors. At q a primitive 15th or 18th root of unity these sums establish recoupling relations on trivalent ribbon graphs labeled by projectors which we use to construct a Witten-Reshetikhin-Turaev invariant of 3-manifolds for Uq( g2 ). The recoupling relations are also sufficient to show that the quotient categories at primitive 15th or 18th roots of unity are semisimple tensor categories with finite number of simple objects. Finally, we compute some examples, including a formula for determining the invariant on all lens spaces. In particular, we show that the invariant is able to differentiate non-homeomorphic 3-manifolds which are homotopically equivalent. |
