Example Computation of hat version of Heegaard Floer Homology

A simple example is ${\mathbb{S}}^3$, shown below.

Figure: Pointed generic Heegaard diagram of ${\mathbb{S}}^3$.

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Let $P$ denote a periodic domain.
The following table lists the connecting domains $A\in {\hat{\pi }}_2(x,y)$.

Table: Table of the connecting domain of the diagram.

$y\backslash x$	$a_1{b}_2$	$a_1{b}_3$	$a_2{b}_2$	$a_2{b}_3$	$a_3{b}_1$
$a_1{b}_2$	$P$	$-D_2$	$D_1$	$D_1-D_2$	$D_1-D_3$
$a_1{b}_3$	$D_2$	$P$	$D_1+D_2$	$D_1$	$D_1+D_2-D_3$
$a_2{b}_2$	$-D_1$	$-D_1-D_2$	$P$	$-D_2$	$-D_3$
$a_2{b}_3$	$-D_1+D_2$	$-D_1$	$D_2$	$P$	$D_2-D_3$
$a_3{b}_1$	$D_3-D_1$	$-D_1-D_2+D_3$	$D_3$	$-D_2+D_3$	$P$

Note

To fill the table, find all bigons or $2n$–gons with Maslov index $1$.
Domains with negative coefficients do not contribute to $\hat{\partial }$
(by the positivity of domains property).

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Hence, the boundary operator matrix is:

$$\hat{\partial }=\left[\begin{array}{ccccc}0& 0& 1& 0& 0\\ 1& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0\end{array}\right].$$

Compute $\ker \hat{\partial }/\mathrm{Im}\hat{\partial }$ to get:

$$\widehat{HF}({\mathbb{S}}^3)={\mathbb{F}}_2.$$

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