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Closing curves by rearranging arcs

Volume 169 / 2022

Leonardo Alese Colloquium Mathematicum 169 (2022), 197-208 MSC: Primary 53A04. DOI: 10.4064/cm8266-6-2021 Published online: 22 February 2022


We show how, under surprisingly weak assumptions, one can split a planar curve into three arcs and rearrange them (matching tangent directions) to obtain a closed curve. We also generalize this construction to curves split into $k$ arcs and comment on what can be achieved by rearranging arcs for a curve in higher dimensions. Proofs involve only tools from elementary topology, and the paper is mostly self-contained.


  • Leonardo AleseDepartment of Mathematics
    TU Graz
    Kopernikusgasse 24
    8010 Graz, Austria

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