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.

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.

## Downloads

## Additional indexing

Item Type: | Dissertation |
---|---|

Referees: | Beliakova A, Schröder V |

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2010 |

Deposited On: | 23 Dec 2010 14:32 |

Last Modified: | 05 Apr 2016 14:28 |

Number of Pages: | 81 |

Related URLs: | http://opac.nebis.ch/F/?local_base=NEBIS&con_lng=GER&func=find-b&find_code=SYS&request=006131192 |

## Download

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.

You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.