Given a pointed Heegaard diagram \begin{align*} ( \Sigma, \vec \alpha, \vec \beta, w) ,\end{align*} the generators of \(\operatorname{CF}\) are formal sums of points \(\mathbf{x} = \sum_{i=1}^{g-k-1} x_i\) where each \(\alpha\) curve contains some \(x_i\) and each \(\beta\) curve contains some \(x_j\), and \(k\) is the number of basepoints in \(\vec w\).

A region is a connected component of the complement of the curves, so \begin{align*} R_i \in \pi_0 \Sigma \setminus\left\{{\vec \alpha, \vec \beta}\right\} .\end{align*} A formal sum of regions is a 2-chain.

Given 2 generators \(\mathbf{x}, \mathbf{y}\) define \(\pi_2(\mathbf{x}, \mathbf{y})\) to be the set of all 2-chains \(\phi\) satisfying \({{\partial}}^2 \phi = \mathbf{y} - \mathbf{x}\). Call such 2-chains domains.

Define \(n_p(\phi)\) to be the coefficient of the region \(R_i \ni p\), then \(\phi\) is a positive domain if \(n_p(\geq 0)\) for all \(p \in \Sigma\setminus\left\{{ \vec \alpha, \vec \beta }\right\}\).

Define \(\pi_2^0(\mathbf{x}, \mathbf{y})\) to be all domains \(\phi\) such that \(n_{\vec w} = 0\). A Heegaard diagram \(\mathcal{H}\) is admissible if for every generator \(\mathbf{x} \in \operatorname{CF}\), every positive domain \(\varphi\in \pi_2^0(\mathbf{x}, \mathbf{x})\) is trivial.

A region is good if it is an \(n{{\hbox{-}}\mathrm{gon}}\) with \(n\leq 4\), and bad if \(n\geq 5\).

For a disc region \(D\), define the badness \begin{align*} b(D) \coloneqq\max\left\{{n-2, 0}\right\} .\end{align*}

Note

• \(D\in 2{{\hbox{-}}\mathrm{gon}}\implies b(D) = 0\)
• \(D\in 4{{\hbox{-}}\mathrm{gon}}\implies b(D) = 0\)
• \(D\in 6{{\hbox{-}}\mathrm{gon}}\implies b(D) = 1\)
• \(D\in 8{{\hbox{-}}\mathrm{gon}}\implies b(D) = 2\)

For a Heegaard diagram \(\mathcal{H}\) define the distance of \(D\) as \begin{align*} d(D) \coloneqq\min \left\{{ {\sharp}(\gamma\cap\vec\beta) {~\mathrel{\Big\vert}~}w'\in D, w \xrightarrow{\gamma} w', \gamma\in \vec\alpha^c }\right\} \in {\mathbf{Z}}_{\geq 0} .\end{align*}

This is the smallest number of intersection points on any arc connecting the basepoint \(w\) to \(w'\in D\). Define the total distance \begin{align*} d( \mathcal{H} ) \coloneqq\max\left\{{ d(D)}\right\} \in {\mathbf{Z}}_{\geq 0} .\end{align*}

For a fixed distance \(d\) define the distance \(d\) complexity as \begin{align*} c_d \coloneqq{\left[ { \sum b(D_i), -b(D_1), -b(D_2), \cdots} \right]} && b(D_1) \geq b(D_2) \geq \cdots \end{align*} For a fixed \(d\), order various \(c_d( \mathcal{H}' )\) lexicographically.