• branched double cover
• branched double covers
• Euler structure
• Euler structures
• knot
• link
• links
• spinC structure
• spinC structures
• Turaev torsion

In recent years some homology theories which are invariants of classical knots in $S^3$ were introduced. The most popular one is Khovanov homology (developed by Khovanov), which has the crucial property that, in a suitable sense, its Euler characteristic is the Jones polynomial. After Khovanov's work, other homology theories were defined, such as odd-Khovanov homology (developed by Ozsv\'ath, Rasmussen and Szab\'o) and knot Floer homology
(defined by Ozsv\'ath and Szab\'o and by Rasmussen).

If $K$ is an alternating knot, all the homology groups named above are `simple' (in the sense that the reduced homology groups are free and supported in one single diagonal with respect to a given bigrading). The knots with `simple' homology groups are called thin knots. Obviously, alternating knots are thin. Manolescu and Ozsv\'ath proved that also quasi-alternating ($\mathcal{QA}$) knots (which are a generalization of alternating knots) are thin. A natural question is whether the converse is true.

Greene proved that the thin knot $10_{50}^n$ is not quasi-alternating. Then, Greene and Watson constructed a family of thin knots $K_n$ such that $K_0=11_{50}^n$ and $K_n$ is not quasi-alternating for $n \gg 0$. Moreover, all the knots $K_n$ have the same homological invariants mentioned above. The aim of this work is to find other infinite families of non-quasi-alternating thin knots with identical homological invariants, using the same techniques used by Greene and Watson. The families that we find are not defined starting from an already known non-quasi-alternating thin knot, so they provide a proof of the existence of non-quasi-alternating thin knots alternative to Greene's counterexample.

The techniques used to obtain this result require a lot of topological tools, introduced in Chapter $1$. After recalling the definitions and the first properties of concepts such as knots and links, handle decompositions, Dehn surgery, other concepts are introduced. First, Spin$^\mathbb{C}$ structures on a manifold are defined, both as isomorphism classes of $\SpinC(n)$-principal bundles as well as elements of the \v Cech cohomology group $\check{\mathrm{H}}^1(\,\cdot\,; C^\infty \SpinC(n))$. The next section deals with the concept of branched cover, with special emphasis on the branched double cover of $S^3$ along a link. It will be proved that, if three links are in a special relation (which is that two of them are the `local resolutions' of the third one), then their branched double covers constitute a triad (which means that they can be obtained from each other by performing certain Dehn surgeries). The last section of Chapter $1$ then gives an overview on the homology theories mentioned above and on a homology theory for $3$-manifolds, called Heegaard Floer homology.

In Chapter $2$ the definition of quasi-alternating link is given, and an obstruction to $\mathcal{QA}$-ness is proved: a lower bound on the correction term of the branched double cover. The correction term, in the cases we are interested in, is closely related to another important invariant of $3$-manifold, which is the Turaev torsion.

The Turaev torsion is defined for an $n$-dimensional manifold endowed with an additional structure, called Euler structure, introduced in the first section of Chapter $3$. In the case of $3$-manifolds there is a canonical identification between Euler structures and Spin$^\mathbb{C}$ structures, as proved in the second section of Chapter $3$. In the last part of Chapter $3$ the Turaev torsion is defined and a way to compute it is presented in the case of $3$-manifolds starting from a cellular decomposition.

In Chapter $4$ several families of knots are introduced. It is proved that knots belonging to the same family have the same homological invariants. Thus, if one knot is thin, so are all the knots belonging to the same family. However, the Turaev torsion of the branched double covers of the knots in a given family is not bounded from below. This implies that also the correction term is unbounded, so, by the obstruction proved in Chapter $2$, infinitely many knots belonging to the family must be non-quasi-alternating. Finally, since this reasoning holds if there exists a thin knot in the family, the last part of Chapter $4$ is devoted to finding such knots.