# Unified quantum SO(3) and SU(2) invariants for rational homology 3-spheres

Bühler, I A. Unified quantum SO(3) and SU(2) invariants for rational homology 3-spheres. 2010, University of Zurich, Faculty of Science.

## Abstract

Inspired by E. Witten’s work, N. Reshetikhin and V. Turaev introduced in 1991 important invariants for 3–manifolds and links in 3–manifolds, the so–called quantum (WRT) SU(2)
invariants. Short after, R. Kirby and P. Melvin defined a modification of these invariants, called the quantum (WRT) SO(3) invariants. Each of these invariants depends on a root of unity.
In this thesis, we give a unification of these invariants. Given a rational homology 3–sphere M and a link L inside, we define the unified invariants ISU(2) M,L and ISO(3)
M,L , 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, i.e. that they lie in modifications of the Habiro ring, which is a cyclotomic completion of the polynomial ring Z[q]. We also give a complete computation of the quantum (WRT) SO(3) and SU(2) invariants of lens spaces with a colored unknot inside.

## Abstract

Inspired by E. Witten’s work, N. Reshetikhin and V. Turaev introduced in 1991 important invariants for 3–manifolds and links in 3–manifolds, the so–called quantum (WRT) SU(2)
invariants. Short after, R. Kirby and P. Melvin defined a modification of these invariants, called the quantum (WRT) SO(3) invariants. Each of these invariants depends on a root of unity.
In this thesis, we give a unification of these invariants. Given a rational homology 3–sphere M and a link L inside, we define the unified invariants ISU(2) M,L and ISO(3)
M,L , 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, i.e. that they lie in modifications of the Habiro ring, which is a cyclotomic completion of the polynomial ring Z[q]. We also give a complete computation of the quantum (WRT) SO(3) and SU(2) invariants of lens spaces with a colored unknot inside.