Witten-Reshetikhin-Turaev Invariants of Seifert Manifolds

摘要

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 2#pi#i/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 Bitten-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.