Volume 597, 2013
What is an Almost Normal Surface?
This paper is dedicated to Hyam Rubinstein on the occasion of his 60th birthday.
Abstract. A major breakthrough in the theory of topological algorithms oc-
curred in 1992 when Hyam Rubinstein introduced the idea of an almost nor-
mal surface. We explain how almost normal surfaces emerged naturally from
the study of geodesics and minimal surfaces. Patterns of stable and unstable
geodesics can be used to characterize the 2-sphere among surfaces, and similar
patterns of normal and almost normal surfaces led Rubinstein to an algorithm
for recognizing the 3-sphere.
1. Normal Surfaces and Algorithms
There is a long history of interaction between low-dimensional topology and the
theory of algorithms. In 1910 Dehn posed the problem of finding an algorithm to
recognize the unknot . Dehn’s approach was to check whether the fundamental
group of the complement of the knot, for which a finite presentation can easily
be computed, is infinite cyclic. This led Dehn to pose some of the first decision
problems in group theory, including asking for an algorithm to decide if a finitely
presented group is infinite cyclic. It was shown about fifty years later that general
group theory decision problems of this type are not decidable .
Normal surfaces were introduced by Kneser as a tool to describe and enumer-
ate surfaces in a triangulated 3-manifold . While a general surface inside a
3-dimensional manifold M can be floppy, and have fingers and filligrees that wan-
der around the manifold, the structure of a normal surface is locally restricted.
When viewed from within a single tetrahedron, normal surfaces look much like flat
planes. As with flat planes, they cross tetrahedra in collections of triangles and
quadrilaterals. Each tetrahedron has seven types of elementary disks of this type;
four types of triangles and three types of quadrilaterals. The whole manifold has
7t elementary disk types, where t is the number of 3-simplices in a triangulation.
Kneser realized that the local rigidity of normal surfaces leads to finiteness
results, and through them to the Prime Decomposition Theorem for a 3-manifold.
This theorem states that a 3-manifold can be cut open along finitely many 2-spheres
into pieces that are irreducible, after which the manifold cannot be cut further in
a non-trivial way. The idea behind this theorem is intuitively quite simple: if a
2010 Mathematics Subject Classification. Primary 57N10; Secondary 53A10.
Key words and phrases. Almost normal surface, minimal surface, 3-sphere recognition.
Partially supported by NSF grant IIS 1117663.
c 2013 American Mathematical Society