Geodesic
Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 1, pp. 95-149

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Each convex smooth curve on the plane has at least four points at which the curvature of the curve has local extrema. If the curve is generic, then it has an equidistant curve with at least four cusps. Using the language of contact topology, V. I. Arnol'd formulated conjectures generalizing these classical results to co-oriented fronts on the plane, namely, the four-vertex conjecture and the four-cusp conjecture. In the present paper these conjectures and some related results are proved. Along with a simple generalization of the Sturm–Hurwitz theory, the main ingredient of the proof is a theory of pseudo-involutions which is constructed in the paper. This theory describes the combinatorial structure of fronts on a cylinder. Also discussed is the relationship between the theory of pseudo-involutions and bifurcations of Morse complexes in one-parameter families. 
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     author = {P. E. Pushkar' and Yu. V. Chekanov},
     title = {Combinatorics of fronts of {Legendrian} links and the {Arnol'd} 4-conjectures},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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P. E. Pushkar'; Yu. V. Chekanov. Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 1, pp. 95-149. http://geodesic.mathdoc.fr/item/RM_2005_60_1_a2/