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Non Semi-Simple ${\mathfrak {sl}(2)}$ Quantum Invariants, Spin Case
Christian Blanchet
,
Francesco Costantino
,
Nathan Geer
,
Bertrand Patureau-Mirand
Abstract
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum ${\mathfrak {sl}(2)}$ were obtained by the last three authors in Costantino et al. (To appear in J. Topology. arXiv:1202.3553). In their construction, the quantum parameter $q$ is a root of unity of order $2r$ where $r > 1$ is odd or congruent to 2 modulo 4. In this paper, we consider the remaining cases where $r$ is congruent to zero modulo 4 and produce invariants of 3-manifolds with colored links, equipped with generalized spin structure. For a given 3-manifold $M$, the relevant generalized spin structures are (non canonically) parametrized by $H^{1}(M;\mathbb{C}/2\mathbb{Z})$.