Upsilon type invariants from link Floer homology

We recall the construction of the upsilon family of concordance invariants of knots and links in the 3-sphere. In the case of knots such invariants were first introduced by Ozsvath, Stipsicz and Szabo in the form of a real-valued function on the interval [0,2]. In general, using results of Alfieri and Livingston, we extend our family to be parametrized by south-west regions of the plane. Moreover, we describe a different proof of the concordance inavariance, using grid diagrams, that allows us to prove this property also when working with links; we then conclude by discussing about the corresponding lower bound for the slice genus.