# Resolving a mismatch in indexing conventions of knot/link Floer homologies

I have trouble matching the indexing conventions for Ozsvath-Szabo's knot Floer homology with link Floer homology.

Say we have a knot $K$ in a 3-sphere. Then we can consider the filtered chain complex $\widehat{CFK}(K)$ of the knot, in which the filtration level is indexed by integers so that the $i$th associated graded complex has homology group $\widehat{HFK}(K,i)$. In link Floer homology, they use the notation $\widehat{CFL}$, but the two notions coincide in case of knots, so I do not distinguish them here. The indexing set is parametrizing the relative spin$^c$ structure $Spin^c(S^3,K)$, which has an affine $\mathbb{Z}$-structure, so that

1. In knot Floer homology case, the spin$^c$ structure $\mathfrak{t}_i$ corresponding to the integer $i$ is uniquely determined by the condition $\left<c_1(\mathfrak{t}_i),[\hat{F}]\right>=2i$, where we identify $Spin^c(S^3,K)$ with $Spin^c(S^3_0(K))$ and $\hat{F}$ is the capping-off of the Seifert surface $F$ of the knot $K$ inside the 0-surgery $S^3_0(K)$. Or equivalently, $c_1(\mathfrak{t}_i)=2i\cdot PD[\mu]$ holds where $\mu$ is the meridian of $K$. See subsection 3.4 in the first paper.
2. In link Floer homology, the spin$^c$ structure corresponding to $i$ is uniquely determined by the condition $c_1(\mathfrak{t}_i)-PD[\mu]=2i\cdot PD[\mu]$, from the sentence below Definition 4.9 in the second paper.

...and a mismatch.

I know, maybe one would not really care about matching the conventions. But considering the symmetry properties of knot Floer homology, which is of course what we don't ignore in computations, there is a reason one wants the right convention to use in practice. Moreover, it seems the second convention does not even make sense, as $c_1$ must be even in $H^2(S^3,K)$. So, a question:

Am I misunderstanding something? Or is this indeed a mismatch?

• I think that Section 8 in the HFL paper should address your concerns. I'm still wrapping my head around it, but it looks like the key idea is that $\mathbb{H} \cong \mathbb{Z}$ in HFL should not be thought of as being the same as $\mathbb{Z}$ in HFK. Apr 28, 2018 at 8:25
• @MarcoGolla I also have Section 8 in HFL paper in mind. Especially they made the identification $s\cdot [\mu]=s$ there. In fact what makes me doubt about the notation in HFL paper is the weird appearance of additional $[\mu]$ summand. This summand must force the affine space $\mathbb{H}$ to be supported on half-integers, not the integers. Apr 28, 2018 at 19:12