Closed planar curves without inflections

Authors:
Shuntaro Ohno, Tetsuya Ozawa and Masaaki Umehara

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

MSC (2010):
Primary 53A04; Secondary 53A15, 53C42

Published electronically:
June 1, 2012

MathSciNet review:
2996970

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Abstract | References | Similar Articles | Additional Information

Abstract: We define a computable topological invariant for generic closed planar regular curves , 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 of a given .

**1.**V. I. Arnol′d,*Topological invariants of plane curves and caustics*, University Lecture Series, vol. 5, American Mathematical Society, Providence, RI, 1994. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. MR**1286249****2.**Grant Cairns and Daniel M. Elton,*The planarity problem for signed Gauss words*, J. Knot Theory Ramifications**2**(1993), no. 4, 359–367. MR**1247573**, 10.1142/S0218216593000209**3.**Fr. Fabricius-Bjerre,*On the double tangents of plane closed curves*, Math. Scand**11**(1962), 113–116. MR**0161231****4.**J. Scott Carter,*Classifying immersed curves*, Proc. Amer. Math. Soc.**111**(1991), no. 1, 281–287. MR**1043406**, 10.1090/S0002-9939-1991-1043406-7**5.**Benjamin Halpern,*Global theorems for closed plane curves*, Bull. Amer. Math. Soc.**76**(1970), 96–100. MR**0262936**, 10.1090/S0002-9904-1970-12380-1**6.**Benjamin Halpern,*An inequality for double tangents*, Proc. Amer. Math. Soc.**76**(1979), no. 1, 133–139. MR**534404**, 10.1090/S0002-9939-1979-0534404-2**7.**Osamu Kobayashi and Masaaki Umehara,*Geometry of scrolls*, Osaka J. Math.**33**(1996), no. 2, 441–473. MR**1416058****8.**Tetsuya Ozawa,*On Halpern’s conjecture for closed plane curves*, Proc. Amer. Math. Soc.**92**(1984), no. 4, 554–560. MR**760945**, 10.1090/S0002-9939-1984-0760945-0**9.**T. Ozawa, Topology of Planar Figures, Lecture Notes in Math. 9, Bifukan Inc. (in Japanese)**92**(1997).**10.**Gudlaugur Thorbergsson and Masaaki Umehara,*Inflection points and double tangents on anti-convex curves in the real projective plane*, Tohoku Math. J. (2)**60**(2008), no. 2, 149–181. MR**2428859****11.**Masaaki Umehara,*6-vertex theorem for closed planar curve which bounds an immersed surface with nonzero genus*, Nagoya Math. J.**134**(1994), 75–89. MR**1280654****12.**Hassler Whitney,*On regular closed curves in the plane*, Compositio Math.**4**(1937), 276–284. MR**1556973**

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Additional Information

**Shuntaro Ohno**

Affiliation:
Department of Mathematics, Kyoto Koka Senior High School, Nishi-kyogoku Nodacho, Kyoto 615-0861, Japan

Email:
tkmskr0329@yahoo.co.jp

**Tetsuya Ozawa**

Affiliation:
Department of Mathematics, Meijo University, Tempaku, Nagoya, 468-8502 Japan

Email:
ozawa@meijo-u.ac.jp

**Masaaki Umehara**

Affiliation:
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-34, O-okayama Meguro-ku, Tokyo 152-8552, Japan

Email:
umehara@is.titech.ac.jp

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

Keywords:
Plane curve,
inflection point,
double tangent

Received by editor(s):
December 21, 2010

Received by editor(s) in revised form:
June 21, 2011, and June 26, 2011

Published electronically:
June 1, 2012

Additional Notes:
The third author was partially supported by the Grant-in-Aid for Scientific Research (A) No. 22244006, Japan Society for the Promotion of Science.

Communicated by:
Daniel Ruberman

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.