$\Sigma$, we mean the image of an embedding $S^1 \hookrightarrow \Sigma$.

A *Heegaard diagram* is a triple $D=(\Sigma,{\mathcal{U}},{\mathcal{L}})$ where

-   $\Sigma$ is a closed, connected, and oriented surface of genus $g\geq 1$,

-   ${\mathcal{U}}$ is a set of $g$ pairwise disjoint circles on $\Sigma$ such that $\Sigma \setminus \cup_{u \in {\mathcal{U}}} u$ is connected,

-   ${\mathcal{L}}$ is a set of $g$ pairwise disjoint circles on $\Sigma$ such that $\Sigma \setminus \cup_{l \in {\mathcal{L}}} l$ is connected,

-   each $u \in {\mathcal{U}}$ is transverse to each $l \in {\mathcal{L}}$.

The circles in ${\mathcal{U}}$ (resp. ${\mathcal{L}}$) are called the *upper* (resp. *lower*) *circles* of the diagram. The last condition implies that the set $(\cup_{u \in {\mathcal{U}}} u) \cap (\cup_{l \in {\mathcal{L}}}l)$ is finite. Its elements are called the *intersection points* of $D$.

For $u \in {\mathcal{U}}$, we denote by $|u|$ the number of intersection points of $D$ lying in $u$. Likewise, for $l \in {\mathcal{L}}$, we denote by $|l|$ the number of intersection points of $D$ lying in $l$. Then the total number of intersection points of $D$ is computed by $$\sum_{u \in {\mathcal{U}}} |u|=\sum_{l \in {\mathcal{L}}} |l|.$$

Heegaard diagrams and 3-manifolds {#secct-Heegaard-diag-3man}
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To any Heegaard diagram is associated a closed 