CEU Electronic Theses and Dissertations, 2018
| Author | Cavallo, Alberto |
|---|---|
| Title | Heegaard Floer Invariants of Smooth and Legendrian Links in Rational Homology 3-spheres |
| Summary | Our main goal in this thesis is to describe the most important results on smooth and Legendrian knots obtained from Heegaard Floer homology in the last ten years and generalize some of them to links. Our first result is a link version of the Thurston-Bennequin inequality. This inequality, in its usual formulation, gives an upper bound for the maximal Thurston-Bennequin and self-linking numbers of a knot $K$ in a tight contact 3-manifolds $(M,\xi)$, in terms of the Euler characteristic of a Seifert surface for $K$. We generalize the inequality to every link $L$, where the resulting upper bound involves the Thurston norm of $L$. This is a rational number extracted from the semi-norm, introduced by Thurston, on the relative second homology group of a 3-manifold with toric boundary. We say that two $n$-component links in $S^3$ are strongly concordant if there is a cobordism between them consisting of $n$ disjoint annuli, each one realizing a knot concordance. Then, starting from a grid diagram $D$ of a link $L$, we define a filtered chain complex $\left(\widehat {GC}(D),\wideha t\partial\right )$ and we prove that its homology, denoted with $\widehat{\mathcal{HFL}}(L)$, is a strong concordance invariant. Since we can prove that $\widehat{\mathcal{HFL}}(L)$ has dimension one in Maslov grading zero, we also extract a numerical invariant from the homology group that we call $\tau(L)$. The $\tau$-invariant gives a lower bound for the slice genus $g_4(L)$, which is the minimum genus of an oriented, compact surface properly embedded in $D^4$ and whose boundary is $L$. We also show that there is a strict relation between the filtration levels of $\widehat{\mathcal{HFL}}(L)$ and the Alexander grading of the torsion-free quotient of $cHFL^-(L)$, a different bigraded version of link Floer homology. Furthermore, we define an invariant of Legendrian links by using open book decompositions. This is done by describing a suitable condition for an open book decomposition $(B,\pi,A)$ to be adapted to a Legendrian link $L$, where $A$ is a system of generators of the first relative homology group of $\overline{\pi^{-1}(1)}$. At this point, we define a special Heegaard diagram $D$ for the link $L$ in the 3-manifold $-M$, given by reversing the orientation on $M$; we have that $D$ is obtained up to isotopy from $(B,\pi,A)$ and then, when $L$ is zero in homology, the invariant $\mathfrak L(L,M,\xi)$ is the isomorphism class of a distinguished cycle $\mathfrak L(D)$ in the link Floer complex $cCFL^-(D,\mathfrak t_{\xi})$. We also give some results on quasi-positive links in $S^3$. In particular, we introduce the subfamily of connected transverse $\C$-links: these are links such that the surface $\Sigma_B$, associated to a quasi-positive braid $B$ for $L$, is connected. We show that, for this kind of links, the slice genus $g_4$ is determined by $\tau$. This allows us to prove that the slice genus is additive under connected sums of connected transverse $\C$-links. |
| Supervisor | Stipsicz, András I. |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2018/cavallo_alberto.pdf |
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