In this article, we use the Lopez-Ros deformation to show that any embedded genus helicoid (or “genus-one helicoid”) must be symmetric with respect to rotation by around a normal line. This partially answers a conjecture of Bobenko. We also show that this symmetry holds for an embedded genus helicoid , provided that the underlying conformal structure of is hyperelliptic.
"Symmetry of embedded genus 1 helicoids." Duke Math. J. 159 (1) 83 - 97, 15 July 2011. https://doi.org/10.1215/00127094-1384791