Relations between Witten-Reshetikhin-Turaev and nonsemisimple sl(2) 3-manifold invariants

摘要

The Witten-Reshetikhin-Turaev (WRT) invariants extend the Jones polynomials of links in S~3 to invariants of links in 3-manifolds. Similarly, the authors constructed two 3-manifold invariants Nr and N_r~0 which extend the Akutsu-Deguchi-Ohtsuki (ADO) invariant of links in S~3 colored by complex numbers to links in arbitrary manifolds. All these invariants are based on the representation theory of the quantum group U_qsl_2, where the definition of the invariants N_r and N_r~0 uses a nonstandard category of U_qsl_2-modules which is not semisimple. In this paper we study the second invariant, N_r~0, and consider its relationship with the WRT invariants. In particular, we show that the ADO invariant of a knot in S~3 is a meromorphic function of its color, and we provide a strong relation between its residues and the colored Jones polynomials of the knot. Then we conjecture a similar relation between N_r~0 and a WRT invariant. We prove this conjecture when the 3-manifold M is not a rational homology sphere, and when M is a rational homology sphere obtained by surgery on a knot in S~3 or a connected sum of such manifolds.