https://neckpinch.com/authors/ellie-decleene/New arXiv papers by Ellie DeCleene, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2403.09876/arxiv:2403.09876Thu, 14 Mar 2024 00:00:00 GMTSigurd Angenent, Evan Patrick Davis, Ellie DeCleene, Paige Ellingson, Ziheng Feng, Edgar Gevorgyan, Aris Lemmenes, Alex Moon, Tyler Joseph Tommasi, Yamin Zhou — 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 ℝ³. 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-Schwarz asymptotic bow-tie result, namely, a vanishing n-loop, when rescaled anisotropically to fit a square bounding box, converges to a "squeezed bow-tie," i.e. the curve \(x, y) : |x|≤ 1, y=± xⁿ⁻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.Curve shorteningSingularitiesNumerics & visualization