Asymptotic expansions of Witten-Reshetikhin-Turaev invariants for some simple 3-manifolds

For any Lie algebra normal letter g and integral level k, there is defined an invariant Z_k* (M, L) of embeddings of links L in 3-manifolds M, known as the Witten-Reshetikhin-Turaev invariant. It is known that for links in S³, Z_k*(S³, L) is a polynomial in q = exp (2πil(k + c_{normal letter g}^υ), namely, the generalized Jones polynomial of the link L. This paper investigates the invariant Z_r-2*(M,Ø) when normal letter g=sl₂ for a simple family of rational homology 3-spheres, obtained by integer surgery around (2, n)-type torus knots. In particular, we find a closed formula for a formal power series Z∞(M) ∈Q[[h]] in h=q-1 from which Z _r-2*(M,Ø) may be derived for all sufficiently large primes r. We show that this formal power series may be viewed as the asymptotic expansion, around q=1, of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition. For these particular manifolds, most of which are not Z-homology spheres, this extends work of Ohtsuki and Murakami in which the existence of power series with rational coefficients related to Z_k*(M, Ø) was demonstrated for rational homology spheres. The coefficients in the formal power series Z∞(M) are expected to be identical to those obtained from a perturbative expansion of the Witten-Chern-Simons path integral formula for Z*(M, Ø).