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

Abstract

For any Lie algebra @Fg 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+c@Fgv), namely, the generalized Jones polynomial of the link L. This paper investigates the invariant Zr−2∗(M,○) when @Fg=@Fs@Fl2 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.