K-noid

Last updated February 06, 2025

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

{\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}}

{\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}}

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

where $_{2}F_{1}(a,b;c;z)$ 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,^[4] k-noids corresponding to the platonic solids and k-noids with handles.^[5]

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References

↑ L. P. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983)
↑ N Schmitt (2007). "Constant Mean Curvature n-noids with Platonic Symmetries". arXiv: math/0702469 .
↑ Matthias Weber (2001). "Classical Minimal Surfaces in Euclidean Space by Examples" (PDF). Indiana.edu. Archived from the original (PDF) on 2019-07-12. Retrieved 2012-10-05.
↑ H. Karcher. "Construction of minimal surfaces, in "Surveys in Geometry", University of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1-96" (PDF). Math.uni-bonn-de. Retrieved 2012-10-05.
↑ Jorgen Berglund, Wayne Rossman (1995). "Minimal Surfaces with Catenoid Ends". Pacific J. Math. 171 (2): 353–371. arXiv: 0804.4203 . Bibcode:2008arXiv0804.4203B. doi:10.2140/pjm.1995.171.353. S2CID 11328539.

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[1] L. P. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983)

[2] N Schmitt (2007). "Constant Mean Curvature n-noids with Platonic Symmetries". arXiv: math/0702469 .

[3] Matthias Weber (2001). "Classical Minimal Surfaces in Euclidean Space by Examples" (PDF). Indiana.edu. Archived from the original (PDF) on 2019-07-12. Retrieved 2012-10-05.

[4] H. Karcher. "Construction of minimal surfaces, in "Surveys in Geometry", University of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1-96" (PDF). Math.uni-bonn-de. Retrieved 2012-10-05.

[5] Jorgen Berglund, Wayne Rossman (1995). "Minimal Surfaces with Catenoid Ends". Pacific J. Math. 171 (2): 353–371. arXiv: 0804.4203 . Bibcode:2008arXiv0804.4203B. doi:10.2140/pjm.1995.171.353. S2CID 11328539.

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