= Relative grading in Heegaard Floer homology
{title2=$\operatorname{gr}(\mathbf x)-\operatorname{gr}(\mathbf y)$}
= Relative Heegaard Floer grading
{synonym}
For generators in the same <Spin-c structure> and a <Whitney disk class> $\psi$ joining them,
$$
\operatorname{gr}(\mathbf x)-\operatorname{gr}(\mathbf y)=\mu(\psi)-2n_z(\psi).
$$
The ambiguity is the divisibility of $c_1(\mathfrak s)$ on $H_2(Y)$; a torsion determinant class gives an integer relative grading. Adding a sphere class changes $\mu$ by two and $n_z$ by one, so it leaves the difference unchanged. For a domain $D$, the index can be calculated as $\mu(D)=e(D)+n_{\mathbf x}(D)+n_{\mathbf y}(D)$, where $e$ is Euler measure and each $n_{\mathbf x}$ is the sum of averages of the four adjacent region multiplicities at the selected intersections.
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