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Jorge-Meeks k-Noid

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Jorge-Meeksk-Noid

A Jorge-Meeks k-noid is a member of the symmetric family of complete minimal surfaces of genus 0 with k catenoidal ends constructed by Jorge and Meeks (1983), where k>=2 is an integer. The Enneper-Weierstrass parameterization can be specified by

f(z)=1/((z^k-1)^2)
(1)
g(z)=z^(k-1).
(2)

The parameter z ranges over the Riemann sphere with the kth roots of unity removed. The total Gaussian curvature is -4pi(k-1) (Jorge and Meeks 1983).

The case k=2 is a catenoid, and k=3 is the trinoid. The cases k=3 and k=4 are illustrated above. For k>=3, the complete surface has self-intersections, although each of its ends is embedded (Meeks and Pérez 2012, p. 20).

See also

Catenoid, Enneper-Weierstrass Parameterization, Minimal Surface, Trinoid

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References

GRAPE. "Jorge-Meeks k-Noids." https://archive.ins.uni-bonn.de/numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/knoid.html.Jorge, L. P. and Meeks, W. H., III. "The Topology of Complete Minimal Surfaces of Finite Total Gaussian Curvature." Topology 22, 203-221, 1983. https://doi.org/10.1016/0040-9383(83)90032-0.Meeks, W. H., III and Pérez, J. A Survey on Classical Minimal Surface Theory. Providence, RI: Amer. Math. Soc., 2012. https://doi.org/10.1090/ulect/060.

Cite this as:

Weisstein, Eric W. "Jorge-Meeks k-Noid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Jorge-Meeksk-Noid.html

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