mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a normal surface is a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

inside a triangulated

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

that intersects each tetrahedron in several components called normal disks. Each normal disk is either a ''triangle'' which cuts off a vertex of the tetrahedron, or a ''quadrilateral'' which separates pairs of vertices. In a given tetrahedron there cannot be two quadrilaterals separating different pairs of vertices, since such quadrilaterals would intersect in a line, causing the surface to be self-intersecting. Dually, a normal surface can be considered as 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 a normal surface can be generalized to arbitrary polyhedra. There are also related notions of almost normal surfaces and spun normal surfaces. In an almost normal surface, one tetrahedron in the triangulation has a single exceptional piece. This is either an ''octagon'' that separates pairs of vertices, or an ''annulus'' that connects two triangles and/or quadrilaterals by a tube. AlmostNormalSurface

The concept of normal surfaces is due to

Hellmuth Kneser Hellmuth Kneser (16 April 1898 – 23 August 1973) was a 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-manifolds. His ...

, who utilized it in his proof of the prime decomposition theorem for 3-manifolds. Later,

Wolfgang Haken Wolfgang Haken (; June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds. Biography Haken was born on June 21, 1928, in Berlin, Germany. His father was Werner Haken, a phys ...

extended and refined the notion to create normal surface theory, which forms the basis of many 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 Univ ...

. The notion of spun normal surface is due to Bill Thurston. Regina is software that enumerates normal and almost-normal surfaces in triangulated 3-manifolds, implementing Rubinstein's 3-sphere recognition algorithm, among other functionalities.

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