Non-semisimple 3-manifold invariants derived from the Kauffman bracket

Abstract

We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of $\mathfrak{s}{\mathfrak{l}}_2$ 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.