connected oriented 3-manifold $M_D$ obtained as follows. Consider the 3-manifold $\Sigma \times [0,1]$. Glue a 2-handle along a tubular neighborhood of each circle $l \times \{0\}$ with $l \in {\mathcal{L}}$. Glue a 2-handle along a tubular neighborhood of each circle $u \times \{1\}$ with $u \in {\mathcal{U}}$. The result is compact connected 3-manifold whose boundary is a number of spheres. We eliminate these spherical boundary components by gluing in 3-balls to obtain $M_D$. We endow $M_D$ with the orientation extending the product orientation of $\Sigma \times [0,1]$.

Any closed connected oriented 3-manifold may be presented this way by a Heegaard diagram. Moreover, the Reidemeister–Singer theorem asserts that two Heegaard diagrams give rise to homeomorphic oriented 3-manifolds if and only if one can be obtained from the other by a finite sequence of four types of moves (and their inverses), namely: a homeomorphism of the surface, an isotopy of the diagram (2-point move), a stabilization, and a handle slide (sliding a circle past another), see Section \[sect-proof-main-thm\] for details.

Oriented, based, and ordered Heegaard diagrams {#sect-opoHD}
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A Heegaard diagram is *oriented* if each lower/upper circle is oriented. A Heegaard diagram is *based* if each lower/upper circle is endowed with a 