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

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For any Lie algebra 𝔤 and integral level k, there is defined an invariant Zk*(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, Zk*(S3, L) is a polynomial in q=exp (2πi/(k+c𝔤v), namely, the generalized Jones polynomial of the link L. This paper investigates the invariant Zr−2*(M,○/) when 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 Zr−2*(M,○/) 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 Zk*(M, ○/) 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, ○/). © 1995 American Institute of Physics.