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Complexity and Quantum Invariants of 3-Manifolds and Knots

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.