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One of the most common methods for representing a closed orientable [[Three-dimensional manifold|three-dimensional manifold]]. A Heegaard diagram of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046800/h0468001.png" /> consists of two systems of simple closed curves on a closed orientable surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046800/h0468002.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046800/h0468003.png" />. The curves of each system satisfy the following conditions: 1) the number of curves in the system is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046800/h0468004.png" />; 2) the curves of the system are disjoint; 3) after cutting the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046800/h0468005.png" /> by these curves, a connected surface must result (a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046800/h0468006.png" /> deleted open discs). Heegaard diagrams are closely connected with Heegaard decompositions (cf. [[Heegaard decomposition|Heegaard decomposition]]): the curves of each system are a complete system of meridians (secants of the circles of the handles) of one handle-body of the decomposition; the curves of the second system are a complete system of meridians of the other handle-body. Two Heegaard diagrams are called equivalent if the Heegaard decompositions corresponding to them are equivalent. It is known, for example, that any two Heegaard diagrams of a three-dimensional sphere are equivalent if they have the same genus. The genus of a Heegaard diagram can always be increased by taking instead of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046800/h0468007.png" /> its connected sum with a two-dimensional torus and by adding to the curves of the diagram the meridian and a longitude of this torus. This operation is called stabilization. Any two Heegaard diagrams of the same manifold are stably equivalent, that is, become equivalent after applying to each of them several stabilization operations.

For references see [[Heegaard decomposition|Heegaard decomposition]].