Tangle functors from semicyclic representations

Abstract

Let q be a 2N th root of unity where N is odd. Let U q (sl 2 ) denote the quantum group with large center corresponding to the lie algebra sl 2 with generators E, F, K, and K −1 . A semicyclic representation of U q (sl 2 ) is an N -dimensional irreducible representation ρ : U q (sl 2 ) → M N (C), so that ρ(E N ) = aId with a = 0, ρ(F N ) = 0 and ρ(K N ) = Id. We construct a tangle functor for framed homogeneous tangles colored with semicyclic representations, and prove that for (1, 1)tangles coming from knots, the invariant defined by the tangle functor coincides with Kashaev's invariant.