mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a normal surface is a surface inside a triangulated

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

that intersects each tetrahedron so that each component of intersection is a ''triangle'' or a ''quad'' (see figure). A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices. A normal surface may have many components of intersection, called normal disks, with one tetrahedron, but no two normal disks can be quads that separate different pairs of vertices since that would lead to the surface 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 Hellmuth Kneser (16 April 1898 – 23 August 1973) was a Baltic German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-mani ...

, 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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

, 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 Joachim Hyam Rubinstein FAA (born 7 March 1948, in Melbourne) an Australian top mathematician specialising in low-dimensional topology; he is currently serving as an honorary professor in the Department of Mathematics and Statistics at the Uni ...

. The notion of spun normal surface is due to

Bill Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. 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.

References

* Hatcher, ''Notes on basic 3-manifold topology''
available online
* Gordon, ed. Kent, ''The theory of normal surfaces''

* Hempel, ''3-manifolds'', American Mathematical Society, * Jaco, ''Lectures on three-manifold topology'', American Mathematical Society, * R. H. Bing, ''The Geometric Topology of 3-Manifolds'', (1983) American Mathematical Society Colloquium Publications Volume 40, Providence RI, .

References

Further reading