Heegaard Splitting

A Heegaard splitting of a connected orientable 3-manifold  is any way of expressing  as the union of two (3,1)-handlebodies along their boundaries. The boundary of such a (3,1)-handlebody is an orientable surface of some genus, which determines the number of handles in the (3,1)-handlebodies. Therefore, the handlebodies involved in a Heegaard splitting are the same, but they may be glued together in a strange way along their boundary. A diagram showing how the gluing is done is known as a Heegaard diagram.

REFERENCES:

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 255, 1994.