#### Volume 15, issue 3 (2015)

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Relations between Witten–Reshetikhin–Turaev and nonsemisimple $\mathfrak{sl}(2)$ $3$–manifold invariants

### Francesco Costantino, Nathan Geer and Bertrand Patureau-Mirand

Algebraic & Geometric Topology 15 (2015) 1363–1386
##### 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.

##### Keywords
quantum invariants, Reshetikhin-Turaev invariants, Hennings invariants, $3$–manifolds
Primary: 57N10
Secondary: 57R56