K-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{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}$$


 * $$\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}$$



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

where $$_2F_1(a,b;c;z)$$ is the Gaussian hypergeometric function and $$\Re \{z\}$$ 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.