k-noid

Trinoid

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.

The term k-noid and trinoid is also sometimes used for constant mean curvature surfaces, especially branched versions of the unduloid ("triunduloids").

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 $f(z)=1/(z^k-1{)}^2,g(z)=z^{k-1}$. This produces the explicit formula

$\begin{array}{cc}X(z)=\frac{1}{2}\mathrm{\Re }\left\{\right(\frac{-1}{kz(z^k-1)}\left)\right[& (k-1)(z^k-1{)}_2{F}_1(1,-1/k;(k-1)/k;z^k)\\ & -(k-1)z^2(z^k-1{)}_2{F}_1(1,1/k;1+1/k;z^k)\\ & -k{z}^k+k+z^2-1\left]\right\}\end{array}$

$\begin{array}{cc}Y(z)=\frac{1}{2}\mathrm{\Re }\left\{\right(\frac{i}{kz(z^k-1)}\left)\right[& (k-1)(z^k-1{)}_2{F}_1(1,-1/k;(k-1)/k;z^k)\\ & +(k-1)z^2(z^k-1{)}_2{F}_1(1,1/k;1+1/k;z^k)\\ & -k{z}^k+k-z^2-1)]\}\end{array}$

$Z(z)=\mathrm{\Re }\left\{\frac{1}{k-k{z}^k}\right\}$

where ${}_2{F}_1(a,b;c;z)$ is the Gaussian hypergeometric function and $\mathrm{\Re }\{z\}$ denotes the real part of $z$.

It is also possible to create k-noids with openings in different directions and sizes, k-noids corresponding to the platonic solids and k-noids with handles.

External links

• Indiana.edu
• Page.mi.fu-berlin.de