Bordered Heegaard Diagram

Definition (Bordered Heegaard diagram).
Given $\mathfrak{Z}$,

$$H=({\Sigma }_g,\underset{{\alpha }^a}{\underbrace{{\alpha }_1^a,\dots ,{\alpha }_{2k}^a}},\underset{{\alpha }^c}{\underbrace{{\alpha }_1^c,\dots ,{\alpha }_{g-k}^c}},\underset{\beta }{\underbrace{{\beta }_1,\dots ,{\beta }_g}},z).$$

Conditions:

• ${\Sigma }_g$: compact genus $g$ surface with one boundary.
• $\beta$: $g$ disjoint circles inside $\Sigma$.
• ${\alpha }^c$: $g-k$ disjoint circles.
• ${\alpha }^a$: $2k$ arcs with endpoints on $\partial \Sigma$.
• $z$: basepoint on $\partial \Sigma \setminus {\alpha }^a$.
• ${\alpha }^a\cap {\alpha }^c=\varnothing$.
• Complements $\Sigma -({\alpha }^a\cup {\alpha }^c)$ and $\Sigma -\beta$ are connected.
• $Z=(\partial \Sigma ,{\alpha }^a\cap \partial \Sigma ,M,z)$ matches endpoints.

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Example.
A Heegaard diagram with $\beta =\{{\beta }_1,\dots ,{\beta }_g\}$ and ${\alpha }^c=\{{\alpha }_1^c,\dots ,{\alpha }_{g-1}^c\}$ represents a 3-manifold with torus boundary. To form a bordered diagram:

1. Choose circles ${\gamma }_1,{\gamma }_2$ with ${\gamma }_1⋔{\gamma }_2=\{p\}$, independent from ${\alpha }^c$.
2. Remove a disk ${\mathbb{D}}^2$ around $p$, disjoint from all other curves.
3. The result is a bordered Heegaard diagram.

Other examples: genus 2 handlebody, trefoil complement.

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Theorem.
Given $\mathcal{H}=(\Sigma ,\alpha ,\beta ,z)$ bordered Heegaard diagram with boundary $\mathfrak{Z}(\mathcal{H})$:

1. Thicken $\Sigma \times [0,1]$.
2. Attach 2-handles along $\alpha$ in $\Sigma \times \{0\}$, and $\beta$ in $\Sigma \times \{1\}$.
3. Boundary parametrization is given by
   $\gamma =({\alpha }^a\cup (\partial \Sigma \setminus nbhd(z)))\times \{0\}$.