Owen Huang Math

I feel like I have learned Heegaard Floer homology like six different times, and every time the algebra is a little different. At first, I was contending with the different algebraic flavours (eg., working over \(WZ=0\) or setting \(W=1\), etc.) But what really annoyed me was translating between the original Ozsvath-Szabo \(\mathbb{F}[U,U^{-1}]\) formulation of the full filtered knot Floer complex \(CFK^{\infty}(K)\) and Zemke’s more recent bigraded \(\mathbb{F}[W,Z]\) formulation. I have procrastinated writing down an explicit translation between these for so long but I’m finally going to put pen to paper (or fingers to keyboard?).

Let me begin by recalling the two conventions and the data they carry.