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 ${\mathfrak{sl}}_2$ 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.

arXiv comment62 pages, Sections 3.2 and 5.1 were added in order to fix a sign problem

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• Diagrammatic Construction of Representations of Small Quantum ${\mathfrak{sl}}_2$