# Upsilon of an alternating knot

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).

The existence of generators $x_i$ and $y_i$ follows from the fact that alternating knots are thin (i.e. their knot Floer homology is supported on a diagonal $M-A=\text{constant}$) and (the filtered quasi-isomorphism class of) their $CFK^\infty$ is determined by the Alexander polynomial. This was proven originally by Ozsváth and Szabó in the paper:

Peter S. Ozsváth and Zoltán Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225–254.

Another, perhaps more modern, exposition is in Petkova's paper:

Ina Petkova, Cables of thin knots and bordered Heegaard Floer homology, Quantum Topol. 4 (2013), no. 4, 377–409.

As for the second question, I'm not sure I understand what you mean by "the role of $y_i$". In my view, they are just generators in the complex with a certain boundary. At a practical level, what they do is they identify each $x_i$ with $U^{i-j}x_j$ in $CFK^\infty$.

You might also want to take a look at Livingston's survey on $\Upsilon$, that has a more transparent reformulation of $\Upsilon$ in terms of certain "slanted filtrations" on $CFK^\infty$.

Charles Livingston, Notes on the knot concordance invariant Upsilon, Algebr. & Geom. Topol. 17 (2017) 111–130.