Normal surface

In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron in several components called normal disks. Each normal disk is a triangle which cuts off a vertex of the tetrahedron, or a quad which separates pairs of vertices. Thus, in a given tetrahedron there cannot be two quads separating different pairs of vertices, since such quads would intersect in a line, meaning the surface would be self-intersecting.



Dually, a normal surface can be considered to be a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner similar to the above.

The concept of normal surface can be generalized to arbitrary polyhedra. There are also related notions of almost normal surface and spun normal surface.

The concept of normal surface is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds. Later Wolfgang Haken extended and refined the notion to create normal surface theory, which is at the basis of many of the algorithms in 3-manifold theory. The notion of almost normal surfaces is due to Hyam Rubinstein. The notion of spun normal surface is due to Bill Thurston.

Regina is software which enumerates normal and almost-normal surfaces in triangulated 3-manifolds, implementing Rubinstein's 3-sphere recognition algorithm, among other things.