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.^[1]

The term k-noid and trinoid is also sometimes used for constant mean curvature surfaces, especially branched versions of the unduloid ("triunduloids").^[2]

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 ${\displaystyle f(z)=1/(z^{k}-1)^{2},g(z)=z^{k-1}\,\!}$.^[3] This produces the explicit formula

${\displaystyle {\begin{aligned}X(z)={\frac {1}{2}}\Re {\Bigg \{}{\Big (}{\frac {-1}{kz(z^{k}-1)}}{\Big )}{\Big [}&(k-1)(z^{k}-1)_{2}F_{1}(1,-1/k;(k-1)/k;z^{k})\\&{}-(k-1)z^{2}(z^{k}-1)_{2}F_{1}(1,1/k;1+1/k;z^{k})\\&{}-kz^{k}+k+z^{2}-1{\Big ]}{\Bigg \}}\end{aligned}}}$

${\displaystyle {\begin{aligned}Y(z)={\frac {1}{2}}\Re {\Bigg \{}{\Big (}{\frac {i}{kz(z^{k}-1)}}{\Big )}{\Big [}&(k-1)(z^{k}-1)_{2}F_{1}(1,-1/k;(k-1)/k;z^{k})\\&{}+(k-1)z^{2}(z^{k}-1)_{2}F_{1}(1,1/k;1+1/k;z^{k})\\&{}-kz^{k}+k-z^{2}-1){\Big ]}{\Bigg \}}\end{aligned}}}$

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

where ${\displaystyle _{2}F_{1}(a,b;c;z)}$ is the Gaussian hypergeometric function and ${\displaystyle \Re \{z\}}$ denotes the real part of ${\displaystyle z}$.

It is also possible to create k-noids with openings in different directions and sizes,^[4] k-noids corresponding to the platonic solids and k-noids with handles.^[5]