Reshetikhin-Turaev invariants of Seifert 3-manifolds and a rational surgery formula

In this paper we construct invariants of 3-manifolds "\`a la Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3-manifold invariants indexed by the integers. We prove this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard Reshetikhin-Turaev-Witten invaria...

We derive formulas for the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra $\mathfrak g$ in terms of the Seifert invariants and standard data for $\mathfrak g$. A main corollary is a determination of the full asymptotic expansions of these invariants for lens spaces in the limit of large quantum level. Our results are in agreement with the asymptotic expansion conjecture due to J. E. Ande...

We report on recent results of the authors concerning calculations of quantum invariants of Seifert 3-manifolds. These results include a derivation of the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated with an arbitrary complex finite dimensional simple Lie algebra, and a determination of the asymptotic expansions of these invariants for lens spaces. Our results are in agreement with the asymptotic expansion conjecture due to JE Andersen [The Witte...

We calculate the large quantum level asymptotic expansion of the RT-invariants associated to SU(2) of all oriented Seifert 3-manifolds X with orientable base or non-orientable base with even genus. Moreover, we identify the Chern-Simons invariants of flat SU(2)-connections on X in the asymptotic formula thereby proving the so-called asymptotic expansion conjecture (AEC) due to J. E. Andersen for these manifolds. For the case of Seifert manifolds with base $S^2$ we actually pr...

We prove that the Witten-Reshetikhin-Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates integral ...

We study the relations between the invariants $\tau_{RT}$, $\tau_{HKR}$, and $\tau_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko, respectively. In particular, we discuss explicitly how $\tau_L$ specializes to $\tau_{RT}$ for semisimple categories and to $\tau_{HKR}$ for Tannakian categories. We give arguments for that $\tau_L$ is the most general invariant that stems from an extended TQFT. We introduce a canonical, central element, {\sf Q}, for a qu...

In this article, we obtain an asymptotic expansion formula for the relative Reshetikhin-Turaev invariant in the case that the ambient 3-manifold is gained by doing rational surgery along one component of Whitehead link. In addition, we obtain an asymptotic expansion formula for the Turaev-Viro invariant of the cusped 3-manifold which is gained by doing rational surgery along one component of the Whitehead link.

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold $S^3/\Gamma$ where $\Gamma$ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of the modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral...

$\hat{Z}$ invariants of 3-manifolds were introduced as series in $q=e^{2\pi i\tau}$ in order to categorify Witten-Reshetikhin-Turaev invariants corresponding to $\tau=1/k$. However modularity properties suggest that all roots of unity are on the same footing. The main result of this paper is the expression connecting Reshetikhin-Turaev invariants with $\hat{Z}$ invariants for $\tau\in\mathbb{Q}$. We present the reasoning leading to this conjecture and test it on various 3-man...

We describe the simplest non-trivial modular category $\mathfrak{E}$ with two simple objects. Then we extract from this category the invariant for non-oriented links in 3-sphere and two invariants for 3-manifolds: the complex-valued Turaev - Reshetikhin type invariant $tr_{\varepsilon}$ and the real-valued Turaev - Viro type invariant $tv_{\varepsilon}$. These two invariants for 3-manifolds are related by the equality $|tr_{\varepsilon}|^2\cdot (\varepsilon + 2) = tv_{\vareps...