CO-QUINK · Complexity and Quantum Invariants of 3-Manifolds and Knots
6РП — Действия „Мария Кюри“
- Период
- 2007-05-01 → 2009-04-30
- Финансиране от ЕС
- 136 375 €
- Участници
- 1
- Схема
- IIF
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Накратко на български
Сложността и квантовите инварианти на триизмерни пространства и възли се анализират чрез геометрично разделяне на триъгълници. Тези математически инструменти помагат за по-точното разпознаване и класифициране на сложни пространствени структури.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
Final Activity Report Summary - CO-QUINK (Complexity and Quantum Invariants of 3-Manifolds and Knots)
The focus of the project has been on the notions of complexity and of quantum invariants of 3-manifolds and links. The complexity of a 3-manifold is a certain integer that rigorously expresses an intuitive estimation of how complicated the manifold is. Being a rather natural and powerful invariant, it plays an important role in dealing with one of the central problems of low-dimensional topology, namely the classification of 3-manifolds, in which quantum invariants also play a significant part. A uniting aspect for these two types of invariants is the common framework that is used in the definition of the complexity and in the definition of some (but not all) quantum invariants, namely that of representing manifolds by (possibly ideal) triangulations and certain dual objects called special spines. In the project, we extended this framework to include the case of links in 3-manifolds; the resulting combinatorial presentation of pairs (manifold, link) naturally led to a complexity theory for such pairs, which we developed along the lines of complexity theory for 3-manifolds. Moreover, the same combinatorial setting was used to define, once again in an analogy with the case of 3-manifolds, the so-called Turaev-Viro invariants of colored links in arbitrary 3-manifolds, and it is the study of the latter that can be regarded as the main achievement of the project. In particular, we formalised the (already existing) definition of Turaev-Viro invariants of coloured links on the basis of certain abstract initial data, that are collections of objects which may come, and in all known cases do, from a modular category, and we established some basic properties of these invariants, such as their behaviour under connected sums. Lastly, we considered the relations of these invariants with other known invariants of links such as the HOMFLY polynomial, the Kauffman polynomial, and the Alexander polynomials (proving that the Turaev-Viro invariants are independent from all of them), as well as with other quantum invariants, most notably with the Witten-Reshetikhin-Turaev ones.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
The complexity of a 3-manifold is a certain integer that rigorously expresses an intuitive estimation of how complicated the manifold is. Being a rather natural and powerful invariant (for instance, only finitely many closed irreducible manifolds can share a given complexity), it plays an important role in dealing with one of the central problems of low-dimensional topology, namely the classification of 3-manifolds. However, its exact position with respect to the rest of 3-dimensional topology, and particularly its connections with other 3-manifold invariants, are not fully understood yet. Only relations with homology and the hyperbolic volume have been explicitly described so far.The aim of the project is to fill this gap (at least to some extent), by establishing connections of complexity with invariants other than homology and the volume, and by exploring several new approaches to complexity. In particular, we expect to discover relations with the theory of quantum invariants, which play an especially important role in the above-mentioned classification task. To meet this goal, we plan to found a certain new combinatorial technique for treating framed links in 3-manifolds. Since surgery on framed links has a crucial importance within the theory of quantum invariants, we believe that this technique will also be a helpful tool for studying these invariants for their own sake.Furthermore, we would like to develop the theory of complexity in several other directions, specifically to analyse its connection with domination between 3-manifolds and to investigate certain notions of complexity for 3-orbifolds and knots. Meeting these aims will use, among other tools, methods relevant to other branches of mathematics, such as geometric group theory and the theory o f accessibility in groups. The foundation of these methods that is foreseen in the course of the project can be expected to make a useful contribution to these areas as well.
Оригинален текст от CORDIS (на английски).
Участници
- UNIVERSITA DI PISA · PISAКоординаторИталия
Връзки
Данни: CORDIS, © Европейски съюз