Unified Quantum SO(3) and SU(2) Invariants for Rational Homology 3-Spheres

TL;DR

This thesis introduces unified quantum invariants for rational homology 3-spheres that encompass both SO(3) and SU(2) WRT invariants, demonstrating their integrality and broad applicability.

Contribution

It unifies quantum WRT invariants for rational homology 3-spheres, covering both SO(3) and SU(2) cases, with strong integrality properties.

Findings

Unified invariants evaluate to quantum WRT invariants at roots of unity.

Invariants dominate all SO(3) quantum WRT invariants.

For odd order first homology, they encompass all SU(2) invariants.

Abstract

In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the corresponding quantum WRT invariant. In the SU(2) case, we assume the order of the first homology group of the manifold to be odd. Therefore, for rational homology 3-spheres, our invariants dominate the whole set of SO(3) quantum WRT invariants and, for manifolds with the order of the first homology group odd, the whole set of SU(2) quantum WRT invariants. We further show, that the unified invariants have a strong integrality property.