## Papers

**Rational cobordisms and integral homology**

(*joint work with Paolo Aceto and JungHwan Park*)

[Abstract]
[pdf]
Preprint (2018).

We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by aunique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of 2-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.
**Upsilon invariants from cyclic branched covers
**

(*joint work with Antonio Alfieri and Andras Stipsicz*)

[Abstract]
[pdf]
Preprint (2018).

We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched

covers with m a prime power, this extension provides new knot concordance invariants $\Upsilon_m^C(K)$. We give computations of these invariants for some

families of alternating knots and reprove some independence results.
**Heegaard Floer homology and concordance bounds on the Thurston norm
**

(*joint work with Marco Golla*)

[Abstract]
[pdf]
Preprint (2018).

We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes

determined by the strong concordance class of a 2-component link L in S^3. We then specialise this procedure to knots in S^2 x S^1, and

obtain a lower bound on their geometric winding number. Furthermore we produce an obstruction for a knot in S^3 to have untwisting number 1.

We then provide an infinite family of null-homologous knots with increasing geometric winding number, on which the bound is sharp.
**The Reidemeister graph is a complete knot invariant**

(*joint work with Agnese Barbensi*)

[Abstract]
[pdf]
Submitted (2018).

We describe two locally finite graphs naturally associated to each knot type K, called Reidemeister graphs. We determine several local and global properties

of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly, we introduce another object,

relating the Reidemeister and Gordian graphs, and determine some of its properties.
**On concordances in 3-manifolds**

[Abstract]
[pdf]
Journal of Topology 11.1 (2018).

We describe an action of the concordance group of knots in S^3 on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of

almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manifolds.

Afterwards, we focus the attention on the case of lens spaces, and use a modified version of the Ozsváth-Szabó-Rasmussen's tau-invariant to obstruct

almost-concordances and prove that each L(p,1) admits infinitely many nullhomologous non almost-concordant knots.

Finally we prove an inequality involving the cobordism PL-genus of a knot and its tau-invariants, in the spirit of Sarkar's cobordims genus bounds.
**A note on cobordisms of algebraic knots **

(*joint work with József Bodnár and Marco Golla*)

[Abstract]
[pdf]
AGT., vol 17 n.4 (2017).

In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves.

The main tool will be the concordance invariant $\nu^+$: we study its behaviour with respect to connected sums, providing an explicit formula in the

case of L-space knots and proving subadditivity in general.
**Cuspidal curves and Heegaard Floer homology**

(*joint work with József Bodnár and Marco Golla*)

[Abstract]
[pdf]
Proc. London Math. Soc. (2016) 112 (3).

We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston.

We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity,

the genus, and the degree of the curve; we improve on some degree-multiplicity asymptotic inequalities; finally, we prove some finiteness results,

we construct infinite families of examples, and in some cases we give an almost complete classification.
**Grid homology in lens spaces**

[Abstract]
[pdf]
Ph.D. Thesis (2016).

I defended my Thesis on the 18th of March at the University of Florence.
**Grid homology in lens spaces with integer coefficients**

[Abstract]
[pdf]
arXiv preprint (2015).

We present a combinatorial proof for the existence of the sign refined Grid Homology in lens spaces, and a self contained proof that $\partial^2_{\mathbb{Z}} =0$.

We also present a Sage program that computes $\widehat{GH}(L(p,q),K;\mathbb{Z})$, and provide empirical evidence supporting the absence

of torsion in these groups.