base point distinct from the intersection points. A Heegaard diagram $D=(\Sigma,{\mathcal{U}},{\mathcal{L}})$ is *ordered* if the sets ${\mathcal{U}}$ and ${\mathcal{L}}$ are (fully) ordered.

The intersection points of an oriented Heegaard diagram $D=(\Sigma,{\mathcal{U}},{\mathcal{L}})$ have a sign defined as follows. Let $c$ be an intersection point between an upper circle $u$ and a lower circle $l$. Then $c$ is *positive* if $(d_cu,d_c l)$ is a positively-oriented basis of $\Sigma$ and *negative* otherwise.

Let $D=(\Sigma,{\mathcal{U}},{\mathcal{L}})$ be an ordered oriented based Heegaard diagram. Then the set ${\mathcal{I}}$ of intersection points of $D$ inherits a total order from ${\mathcal{U}}$ as follows. For $u \in {\mathcal{U}}$, denote by ${\mathcal{I}}_u$ the set of intersection points of $D$ lying in $u$. Enumerating the elements of ${\mathcal{I}}_u$ starting from the base point of $u$ and following the orientation of $u$ defines a total order on ${\mathcal{I}}_u$. Then ${\mathcal{I}}=\amalg_{u \in {\mathcal{U}}} {\mathcal{I}}_u$ is endowed with the lexicographic order induced by the order of ${\mathcal{U}}$ and that of ${\mathcal{I}}_u$. In other words, for $c \in {\mathcal{I}}_u$ and $c' \in {\mathcal{I}}_{u'}$ with $u,u'\in{\mathcal{U}}$, we let $c \leq c'$ if $u < u'$ in ${\mathcal{U}}$ or if $u=u'$ and $c \leq c'$ in ${\mathcal{I}}_u$. Similarly, replacing ${\mathcal{U}}$ with ${\mathcal{L}}$, we obtain another total order 