Closed planar curves without inflections

Ohno, Shuntaro; Ozawa, Tetsuya; Umehara, Masaaki

doi:10.1090/S0002-9939-2012-11319-X

Closed planar curves without inflections
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by Shuntaro Ohno, Tetsuya Ozawa and Masaaki Umehara

Proc. Amer. Math. Soc. 141 (2013), 651-665

DOI: https://doi.org/10.1090/S0002-9939-2012-11319-X

Published electronically: June 1, 2012

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Abstract:

We define a computable topological invariant $\mu (\gamma )$ for generic closed planar regular curves $\gamma$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant $\mu (\gamma )$ of a given $\gamma$.

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