  We prove that a closed immersed plane curve with total curvature $2\pi m$ has
entropy at least $m$ times the entropy of the embedded circle, as long as it
generates a type I singularity under the curve shortening flow (CSF). We
construct closed immersed plane curves of total curvature $2\pi m$ whose
entropy is less than $m$ times the entropy of the embedded circle. As an
application, we extend Colding-Minicozzi's notion of a generic mean curvature
flow to closed immersed plane curves by constructing a piecewise CSF whose only
singularities are embedded circles and type II singularities.
