A Heegaard splitting divides a closed three-manifold into two handlebodies with common boundary. For a three-manifold with boundary, one uses compression bodies, which can retain negative boundary components.
A compression body is obtained from a product by attaching one-handles to , allowing three-ball components as well. Its negative boundary is the retained copy of and its positive boundary is the other boundary component. A handlebody has empty negative boundary.
A Heegaard diagram marks the boundaries of compressing disks from the two sides of a Heegaard surface. Attaching two-handles along the two collections, and capping resulting spherical boundary components, reconstructs the three-manifold.
Thicken the planar graph underlying a knot diagram. Its boundary supports a Heegaard diagram with face disks on one side and crossing-tunnel disks on the other. The face generators and crossing curves give a Dehn presentation of a knot group.
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Heegaard splitting is a concept from the field of topology, specifically in the study of 3-manifolds. It provides a way to understand the structure of a 3-manifold by decomposing it into simpler pieces. The key idea revolves around the partitioning of a 3-manifold into two "handlebodies.