Relations between Witten–Reshetikhin–Turaev and nonsemisimple sl.2/ 3–manifold invariants

Abstract

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 N r and N 0 r 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 q sl 2, where the definition of the invariants N r and N 0 r uses a nonstandard category of U q sl 2 –modules which is not semisimple. In this paper we study the second invariant, N 0 r, 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 0 r 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.