Non-Semisimple 3-Manifold Invariants Derived From the Kauffman Bracket

We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of $\mathfrak{sl}_2$ using purely combinatorial methods based on Temperley-Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten-Reshetikhin-Turaev invariants, which can be reformulated following our approach in the case of rational homology spheres.
Comment: 62 pages, Sections 3.2 and 5.1 were added in order to fix a sign problem