Exponential mechanism (differential privacy): Difference between revisions

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In [[mathematics]], the '''Gieseking manifold''' is a [[hyperbolic 3-manifold|cusped hyperbolic]] [[3-manifold]] of finite volume. It is [[orientability|non-orientable]] and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by {{harvtxt|Gieseking|1912}}.
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The Gieseking manifold can be constructed by removing the vertices from a [[tetrahedron]], then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3.  Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order.  Glue the face 0,2,3 to the face 3,2,1 in that order.   In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of Epstein-Penner.  Moreover, the angle made by the faces is <math>\pi/3</math>. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.  
 
The Gieseking manifold has a [[covering space|double cover]] [[homeomorphism|homeomorphic]] to the [[Figure eight knot (mathematics)|figure eight knot]] [[knot complement|complement]]. The underlying compact manifold has boundary a [[Klein bottle]], and the first homology group of the Gieseking manifold is the integers.
 
The Gieseking manifold is a fiber bundle over the circle with fiber the once-punctured torus and monodromy the [[Arnold's cat map]].
 
== References ==
 
* {{citation|first=H. |last=Gieseking |title=Analytische Untersuchungen über Topologische Gruppen|series= Thesis, Muenster |year=1912| jfm =43.0202.03 |url=http://name.umdl.umich.edu/ABR1814.0001.001 }}
*{{Citation | last1=Adams | first1=Colin C. | title=The noncompact hyperbolic 3-manifold of minimal volume | doi=10.2307/2046691 | mr=894423 | year=1987 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=100 | issue=4 | pages=601–606}}
 
[[Category:Geometric topology]]
[[Category:Hyperbolic geometry]]
[[Category:3-manifolds]]

Latest revision as of 03:43, 11 November 2014

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