## Abstract

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 S^{3}, Z_{k}*(S^{3}, 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=sl_{2} 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).

Original language | American English |
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Pages (from-to) | 6106-6129 |

Number of pages | 24 |

Journal | Journal of Mathematical Physics |

Volume | 36 |

Issue number | 11 |

DOIs | |

State | Published - 1995 |

Externally published | Yes |