The Helicoid with Handle(s)

Hoffman, Karcher and Wei added handles to the helicoid. All the pictures on this page are made using VRML models of the surfaces by J. Hoffman.
The most simple of these surfaces is the singly periodic helicoid which is invariant under a translation so that the quotient surface has genus one (compare this to Riemann's minimal surface). It contains several straight lines: There is a horizontal line which is invariant under the translation and there are also two further lines within a fundamental piece hich are orthogonal to the horizontal line. Singly periodic helicoid with handles
This surface admits a twist deformation. The twist angle is the angle by which the surface normal turns around when moving along the horizontal straight line in a fundamental piece. The twist angle can be less then 2 pi, squeezing the surface.More importantly, it can be arbitrarily large.
Twist below 2 pi twist above 2 pi
For infinite twist angle, these surfaces converge to the genus one helicoid. This is the first example of an infinite total curvature but finite topology embedded minimal surface.
Genus one helicoid