Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories

In this paper, we construct invariants of 3-manifolds ‘à la Reshetikhin–Turaev ’ in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories that lead to a family of 3-manifold invariants indexed by the integers. We prove that 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 Witten–Reshetikhin–Turaev invariants do not and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infinite class of links.