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

**R. Lawrence**

**Abstract: **For any Lie algebra, *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{2\pi{}i\over(k+2)}*, namely the generalised Jones polynomial
of the link *L*. This paper investigates the invariant
*Z^*_{r-2}(M,\emptyset)* when *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_\infty(M)\in***Z**[[*h*]] in
*h*=*q*-1 from which *Z^*_{r-2}(M,\emptyset)* 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
multi-valued 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 T. Ohtsuki
and H. Murakami in which the existence of power series with rational
coefficients related to *Z^*_k(M,\emptyset)* was demonstrated for
integral homology spheres. The coefficients in the formal power series
*Z_\infty(M)* are expected to be identical to those obtained from a
perturbative expansion of the Witten-Chern-Simons path integral formula for
*Z^*(M,\emptyset)*.

**Keywords: **knot theory, manifold invariants, Feynman integrals,
p-adic convergence, topological field theory

**AMS subject classification: **57N10 (11B68 57M25 58G18)

**Length: **24 pages

**Reference: ****Invited contribution** for the Special Issue on
`Quantum geometry and diffeomorphism invariant quantum field theory' *
J. Math. Phys. ***36 ** (1995) 6106-6129.
MR1355900 (97a:57015) (review by * Louis H. Kauffman *.)

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*Last updated on September 4th, 1996.*

ruthel@ma.huji.ac.il