  We show that for certain one-parameter families of initial conditions in
$\mathbb R^3$, when we run mean curvature flow, a genus one singularity must
appear in one of the flows. Moreover, such a singularity is robust under
perturbation of the family of initial conditions. This contrasts sharply with
the case of just a single flow. As an application, we construct an embedded,
genus one self-shrinker with entropy lower than a shrinking doughnut.
