View source for K-noid

{{DISPLAYTITLE:''k''-noid}}
[[File:Trinoid.png|thumb|Trinoid]]
[[File:7-noid.png|thumb|7-noid]]
In [[Differential geometry|differential geometry]], a '''''k''-noid''' is a [[Minimal surface|minimal surface]] with ''k'' [[Catenoid|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 surfaces, especially branched versions of the [[Unduloid|unduloid]] ("triunduloids").<ref>{{cite arXiv|author=N Schmitt|title=Constant Mean Curvature ''n''-noids with Platonic Symmetries|eprint=math/0702469|year=2007}}</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|hypergeometric function]] and <math>\Re \{z\}</math> denotes the real part of <math>z</math>.

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 journal|author=Jorgen Berglund, Wayne Rossman|title= Minimal Surfaces with Catenoid Ends |journal=Pacific J. Math. |volume=171 |issue=2 |year=1995 |pages=353–371|arxiv=0804.4203|bibcode= 2008arXiv0804.4203B |doi=10.2140/pjm.1995.171.353|s2cid= 11328539 }}</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]

{{Minimal surfaces}}

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

{{Sourceattribution|K-noid}}