Heegaard diagram

One of the most common methods for representing a closed orientable three-dimensional manifold. A Heegaard diagram of genus $n$ consists of two systems of simple closed curves on a closed orientable surface $F$ of genus $n$. The curves of each system satisfy the following conditions: 1) the number of curves in the system is $n$; 2) the curves of the system are disjoint; 3) after cutting the surface $F$ by these curves, a connected surface must result (a sphere with $2n$ deleted open discs). Heegaard diagrams are closely connected with Heegaard decompositions (cf. 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 $F$ 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.