Which shapes can appear in a curve shortening flow singularity?*

We study possible tangles that can occur in singularities of solutions to plane curve shortening flow. We exhibit solutions in which more complicated tangles with more than one self-intersection disappear into a singular point. It seems that there are many examples of this kind and that a complete classification presents a problem similar to the problem of classifying all knots in R 3 . As a particular example, we introduce the so-called n -loop curves, which generalize Matt Grayson’s figure-eight curve, and we conjecture a generalization of the Coiculescu–Schwartz asymptotic bow-tie result, namely, a vanishing n -loop, when re-scaled anisotropically to fit a square bounding box, converges to a ‘squeezed bow-tie,’ i.e. the curve { ( x , y ) : | x | ⩽ 1 , y = ± x n − 1 } ∪ { ( ± 1 , y ) : | y | ⩽ 1 } . As evidence in support of the conjecture, we provide a formal asymptotic analysis on one hand, and a numerical simulation for the cases n = 3 and n = 4 on the other.