Relations between Witten–Reshetikhin–Turaev and nonsemisimple $\mathfrak{sl}(2)$ $3$–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 $N_r^0$ 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 $U_q{\mathfrak{s}\mathfrak{l}}_2$, where the definition of the invariants $N_r$ and $N_r^0$ uses a nonstandard category of $U_q{\mathfrak{s}\mathfrak{l}}_2$–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.