I have a couple of questions about how Oz-Stip-Sz computes the upsilon function invariant of an alternating knot in their upsilon ($\Upsilon$) paper here.1 This is theorem 1.14 (on the bottom of page 3); the proof is at the top of page 22. In the proof they state that for alternating knots one can find two sets of generators for $CFK^\infty$ with certain nice properties. These are the $x_i$'s and the $y_j$'s in the proof. My questions are: how do we find the $x_i$ and $y_j$ generators, and what is the role of the $y_j$ generators in determining the upsilon ($\Upsilon$) function?
1Ozsvath, Peter, Andras Stipsicz, and Zoltán Szabó. "Concordance homomorphisms from knot Floer homology." arXiv:1407.1795 (2014).