In mathematics, Fenchel–Nielsen coordinates are coordinates for
Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
introduced by
Werner Fenchel
Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a German-Danish mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear opti ...
and
Jakob Nielsen.
Definition
Suppose that ''S'' is a compact
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
of
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
''g'' > 1. The Fenchel–Nielsen coordinates depend on a choice of 6''g'' − 6 curves on ''S'', as follows. The Riemann surface ''S'' can be divided up into 2''g'' − 2
pairs of pants by cutting along 3''g'' − 3 disjoint simple closed curves. For each of these 3''g'' − 3 curves γ, choose an arc crossing it that ends in other boundary components of the pairs of pants with boundary containing γ.
The Fenchel–Nielsen coordinates for a point of the Teichmüller space of ''S'' consist of 3''g'' − 3 positive real numbers called the lengths and 3''g'' − 3 real numbers called the twists. A point of Teichmüller space is represented by a hyperbolic metric on ''S''.
The lengths of the Fenchel–Nielsen coordinates are the lengths of geodesics homotopic to the 3''g'' − 3 disjoint simple closed curves.
The twists of the Fenchel–Nielsen coordinates are given as follows. There is one twist for each of the 3''g'' − 3 curves crossing one of the 3''g'' − 3 disjoint simple closed curves γ. Each of these is homotopic to a curve that consists of 3 geodesic segments, the middle one of which follows the geodesic of γ. The twist is the (positive or negative) distance the middle segment travels along the geodesic of γ.
See also
*
Dehn twist
In geometric topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area dis ...
References
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Riemann surfaces
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