Integrality and Symmetry of Quantum Link Invariants

0. Introduction. Quantum invariants of framed links whose components are colored by modules of a simple Lie algebra g are Laurent polynomials in v1/D (with integer coefficients), where v is the quantum parameter and D an integer depending on g. We show that quantum invariants, with a suitable normalization, are Laurent polynomials in v2. We also establish two symmetry properties of quantum link invariants at roots of unity. The first asserts that quantum link invariants, at rth roots of unity, are invariant under the action of the affine Weyl group Wr , which acts on the weight lattice. A fundamental domain of Wr is the fundamental alcove C̄r , a simplex. Let G be the center of the corresponding simply connected complex Lie group. There is a natural action of G on C̄r . The second symmetry property, in its simplest form, asserts that quantum link invariants are invariant under the action ofG if the link has zero linking matrix. The second symmetry property generalizes symmetry principles of Kirby and Melvin (the sl2 case) and Kohno and Takata (the sln case) to arbitrary simple Lie algebra.