No abstract

For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl 2 Witten-Reshetikhin-Turaev invariant, Z K , at q = exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten-Chern-Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K −1 which, for K an odd prime power, converges K-adically to the exact total value of the invariant Z K at that root of unity. Evaluations at rational K are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern-Simons-Witten theory is exact for Seifert manifolds and for torus knots in S 3. The possibility of generalising such results is also discussed.

In this paper a topological construction of representations of the series of Hecke algebras, associated with 2-row Young diagrams will be given. This construction gives the representations in terms of the monodromy representation obtained from a vector bundle on which there is a natural flat connection. The fibres of the vector bundle are homology spaces of configuration spaces of points in C, with a suitable twisted local coefficient system. It is also shown that there is a close correspondence between this construction and the work of Tsuchiya and Kanie, who constructed Hecke algebra representations from the monodromy of n-point functions in a conformal field theory on P 1 . This work has significance in relation to the one-variable Jones polynomial, which can be expressed in terms of characters of the Iwahori-Hecke algebras associated with 2-row Young diagrams; it gives rise to a topological description of the Jones polynomial, which will be discussed elsewhere [L2].

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link in S 3 . Our main tool is a careful use of theÅrhus integral and the (now proven) "Wheels" and "Wheeling" conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.