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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

[1] V. I. Arnol'd, Topological Invariants of Plane Curves and Caustics, Univ. Lecture Ser., 5, Amer. Math. Soc., Princeton, RI, 1994 | MR

[2] V. I. Arnold, “Invarianty i perestroiki frontov na ploskosti”, Tr. MIAN, 209, 1995, 14–64 | MR | Zbl

[3] V. I. Arnold, “Topologicheskie problemy teorii rasprostraneniya voln”, UMN, 51:1 (1996), 3–50 | MR | Zbl

[4] V. I. Arnold, Chto takoe matematika?, MTsNMO, M., 2002

[5] V. I. Arnold, A. B. Givental, “Simplekticheskaya geometriya”, Itogi nauki i tekhniki. Ser. Sovrem. problemy matem. Fundam. napravleniya, 4, VINITI, M., 1985, 5–139 | MR

[6] S. A. Barannikov, “The framed Morse complex and its invariats”, Adv. Soviet Math., 21 (1994), 93–115 | MR | Zbl

[7] W. Blaschke, Kreis und Kugel, de Gruyter, Berlin, 1956 ; V. Blyashke, Krug i shar, Nauka, M., 1967 | MR | Zbl | MR

[8] J. Cerf, “Stratification naturalle des espaces de fonctions différentiables réeles et le théorème de la pseudo-isotopie”, Inst. Hautes Études Sci. Publ. Math., 39 (1970), 5–173 | DOI | MR | Zbl

[9] M. Chaperon, “On generating families”, The Floer Memorial Volume, Progr. Math., 133, eds. H. Hofer et al., Birkhäuser, Basel, 1995, 283–296 | MR | Zbl

[10] Yu. V. Chekanov, “Kriticheskie tochki kvazifunktsii i proizvodyaschie semeistva lezhandrovykh mnogoobrazii”, Funkts. analiz i ego pril., 30:2 (1996), 56–69 | MR | Zbl

[11] Yu. V. Chekanov, “Differential algebra of Legendrian links”, Invent. Math., 150:3 (2002), 441–483 | DOI | MR | Zbl

[12] Ya. M. Eliashberg, “Teorema o strukture volnovykh frontov i ee primeneniya v simplekticheskoi topologii”, Funkts. analiz i ego pril., 21:3 (1987), 65–72 | MR

[13] Y. Eliashberg, M. Gromov, “Lagrangian intersections theory: finite-dimensional approach”, Geometry of Differential Equations, Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998, 27–118 | MR | Zbl

[14] M. Entov, “On the necessity of Legendrian fold singularities”, Internat. Math. Res. Notices, 20 (1998), 1055–1077 | DOI | MR | Zbl

[15] E. Ferrand, P. E. Pushkar', “Non-cancellation of cusps on wave fronts”, C. R. Acad. Sci. Paris Sér. I Math., 327:9 (1998), 827–831 | MR | Zbl

[16] D. Fuchs, “Chekanov–Eliashberg invariant of Legendrian knots: existence of augmentations”, J. Geom. Phys., 47:1 (2003), 43–65 | MR | Zbl

[17] D. Fuchs, S. Tabachnikov, “Invariants of Legendrian and transverse knots in the standard contact space”, Topology, 36:5 (1997), 1025–1053 | DOI | MR | Zbl

[18] A. Hatcher, J. Wagoner, “Pseudo-isotopies of compact manifolds”, Astérisque, 6 (1973), 3–274 | MR

[19] A. Hurwitz, “Über die Fourierschen Konstanten integrierbarer Funktionen”, Math. Ann., 57 (1903), 425–446 | DOI | MR | Zbl

[20] S. Mukhopadhyaya, “New methods in the geometry of a plane arc. I: Cyclic and sextactic points”, Bull. Calcutta Math. Soc., 1 (1909), 31–37 | Zbl

[21] P. E. Pushkar, “Obobschenie teoremy Chekanova. Diametry immersirovannykh mnogoobrazii i volnovykh frontov”, Tr. MIAN, 221, 1998, 289–304 | MR | Zbl

[22] M. Schwarz, Morse homology, Progr. Math., 111, Birkhäuser, Basel, 1993 | MR | Zbl

[23] V. D. Sedykh, “Teorema o chetyrekh vershinakh vypukloi prostranstvennoi krivoi”, Funkts. analiz i ego pril., 26:1 (1992), 35–41 | MR | Zbl

[24] D. Théret, “A complete proof of Viterbo's uniqueness theorem on generating functions”, Topology Appl., 96:3 (1999), 249–266 | DOI | MR | Zbl

[25] C. Viterbo, “Symplectic topology as the geometry of generating functions”, Math. Ann., 292:4 (1992), 685–710 | DOI | MR | Zbl