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

Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow