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[[File:Trinoid.png|thumb|Trinoid]]
[[File:7-noid.png|thumb|7-noid]]
In [[differential geometry]], a ''k''-'''noid''' is a [[minimal surface]] with ''k'' [[catenoid]] openings. In particular, the 3-noid is often called trinoid. The first ''k''-noid minimal surfaces were described by Jorge and Meeks in 1983.<ref>L. P. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983)</ref>

The term ''k''-noid and trinoid is also sometimes used for [[constant mean curvature surface]]s, especially branched versions of the [[unduloid]] ("triunduloids").<ref>{{cite web|author=N Schmitt|title=Constant Mean Curvature ''n''-noids with Platonic Symmetries|url=
http://arxiv.org/abs/math/0702469|publisher=Arxiv.org|accessdate=2012-10-05}}</ref>

''k''-noids are topologically equivalent to ''k''-punctured spheres (spheres with ''k'' points removed). ''k''-noids with symmetric openings can be generated using the [[Weierstrass–Enneper parameterization]] <math>f(z) = 1/(z^k-1)^2, g(z) = z^{k-1}\,\!</math>.<ref>{{cite web|author=Matthias Weber|title=Classical Minimal Surfaces in Euclidean Space by Examples|year=2001|url=http://www.indiana.edu/~minimal/research/claynotes.pdf|publisher=Indiana.edu|accessdate=2012-10-05}}</ref> This produces the explicit formula

: <math>\begin{align}
X(z) = \frac{1}{2} \Re \Bigg\{ \Big(\frac{-1}{kz(z^k-1)} \Big) \Big[ &(k-1)(z^k-1)_2F_1(1,-1/k;(k-1)/k;z^k)\\
& {}-(k-1)z^2(z^k-1)_2F_1(1,1/k;1+1/k;z^k) \\
&{}-kz^k +k+z^2-1 \Big] \Bigg\}
\end{align}</math>

: <math>\begin{align}
Y(z) = \frac{1}{2} \Re \Bigg\{ \Big(\frac{i}{kz(z^k-1)}\Big) \Big[ &(k-1)(z^k-1)_2F_1(1,-1/k;(k-1)/k;z^k) \\
&{}+(k-1)z^2(z^k-1)_2F_1(1,1/k;1+1/k;z^k)\\
& {}-kz^k+k-z^2-1 ) \Big] \Bigg\}
\end{align}</math>

: <math>
Z(z) =\Re \left \{ \frac{1}{k-kz^k} \right\}
</math>

where <math>_2F_1(a,b;c;z)</math> is the Gaussian [[hypergeometric function]].

It is also possible to create k-noids with openings in different directions and sizes,<ref>{{cite web|author=H. Karcher|title=Construction of minimal surfaces, in "Surveys in Geometry", University of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1-96|url= http://www.math.uni-bonn.de/people/karcher/karcherTokyo.pdf|publisher=Math.uni-bonn-de|accessdate=2012-10-05}}</ref> k-noids corresponding to the [[platonic solids]] and k-noids with handles.<ref>{{cite web|author=Jorgen Berglund, Wayne Rossman|title= Minimal Surfaces with Catenoid Ends. Pacific J. Math. Volume 171, Number 2 (1995),pp. 353-371|url=http://arxiv.org/abs/0804.4203v1|publisher=Arxiv.org|accessdate=2012-10-05}}</ref>

==References==
{{Reflist}}

==External links==
* [http://www.indiana.edu/~minimal/archive/Spheres/Noids/Jorge-Meeks/web/index.html Indiana.edu]
* [http://page.mi.fu-berlin.de/polthier/booklet/alteration.html Page.mi.fu-berlin.de]

[[Category:Differential geometry]]
[[Category:Minimal surfaces]]

Jacobsthal sum - Revision history

en>Mark viking: Added wl