  We prove: a properly embedded, genus-one minimal surface that is asymptotic
to a helicoid and that contains two straight lines must intersect that helicoid
precisely in those two lines. In particular, the two lines divide the surface
into two connected components that lie on either side of the helicoid. We prove
an analogous result for periodic helicoid-like surfaces. We also give a simple
condition guaranteeing that an immersed minimal surface with finite genus and
bounded curvature is asymptotic to a helicoid at infinity.
