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

TL;DR

This paper introduces a new combinatorial approach to derive non-semisimple quantum invariants of 3-manifolds from the small quantum group of sl2, extending Witten-Reshetikhin-Turaev invariants.

Contribution

It provides a purely combinatorial method using Temperley-Lieb algebras and Kauffman brackets to recover non-semisimple 3-manifold invariants, extending existing quantum invariants.

Findings

Derived invariants for closed oriented 3-manifolds.

Extended Witten-Reshetikhin-Turaev invariants to non-semisimple cases.

Applicable to rational homology spheres.

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.