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

TL;DR

This paper develops new 3-manifold invariants using non-semi-simple categories, which can distinguish manifolds that traditional invariants cannot, and connects these invariants to the Volume Conjecture.

Contribution

It introduces a novel family of 3-manifold invariants derived from non-semi-simple categories, expanding the scope of quantum topology tools.

Findings

Invariants distinguish homotopically equivalent manifolds beyond standard invariants

Invariants can be computed via a set of axioms

A version of the Volume Conjecture is proven for an infinite class of links

Abstract

In this paper we construct invariants of 3-manifolds "\`a la Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3-manifold invariants indexed by the integers. We prove 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 Reshetikhin-Turaev-Witten 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.