Abstract:
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.
Citation:
P. E. Pushkar', Yu. V. Chekanov, “Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures”, Russian Math. Surveys, 60:1 (2005), 95–149
\Bibitem{PusChe05}
\by P.~E.~Pushkar', Yu.~V.~Chekanov
\paper Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures
\jour Russian Math. Surveys
\yr 2005
\vol 60
\issue 1
\pages 95--149
\mathnet{http://mi.mathnet.ru/eng/rm1390}
\crossref{https://doi.org/10.1070/RM2005v060n01ABEH000808}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2145660}
\zmath{https://zbmath.org/?q=an:1085.57008}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2005RuMaS..60...95C}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000229893400003}
\elib{https://elibrary.ru/item.asp?id=25787150}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-20444470598}
Linking options:
https://www.mathnet.ru/eng/rm1390
https://doi.org/10.1070/RM2005v060n01ABEH000808
https://www.mathnet.ru/eng/rm/v60/i1/p99
This publication is cited in the following 58 articles:
LINYI CHEN, GRANT CRIDER-PHILLIPS, BRAEDEN REINOSO, JOSHUA SABLOFF, LEYU YAO, “Non-Orientable Lagrangian Fillings of Legendrian Knots”, Math. Proc. Camb. Phil. Soc., 176:1 (2024), 123
Wenyuan Li, “Existence of generating families on Lagrangian cobordisms”, Math. Ann., 2024
Ivan Dynnikov, Vladimir Shastin, “Distinguishing Legendrian knots with trivial orientation-preserving symmetry group”, Algebr. Geom. Topol., 23:4 (2023), 1849–1889
Vladimir Chernov, Rustam Sadykov, “Conjectures about virtual Legendrian knots and links”, Journal of Geometry and Physics, 193 (2023), 104962
Gil Bor, Serge Tabachnikov, “On Cusps of Caustics by Reflection: Billiard Variations on the Four Vertex Theorem and on Jacobi's Last Geometric Statement”, The American Mathematical Monthly, 130:5 (2023), 454
Petya Pushkar, Misha Temkin, “Enhanced Bruhat Decomposition and Morse Theory”, International Mathematics Research Notices, 2023:19 (2023), 16837
Dylan Cant, “Remarks on the oscillation energy of Legendrian isotopies”, Geom Dedicata, 217:5 (2023)
Dynnikov I., Prasolov M., “Rectangular Diagrams of Surfaces: Distinguishing Legendrian Knots”, J. Topol., 14:3 (2021), 701–860
Haiden F., “Flags and Tangles”, Quantum Topol., 12:3 (2021), 461–505
Le Roux F., Seyfaddini S., Viterbo C., “Barcodes and Area-Preserving Homeomorphisms”, Geom. Topol., 25:6 (2021), 2713–2825
Haiden F., “Legendrian Skein Algebras and Hall Algebras”, Math. Ann., 381:1-2 (2021), 631–684
Sarah Blackwell, Noémie Legout, Caitlin Leverson, Maÿlis Limouzineau, Ziva Myer, Yu Pan, Samantha Pezzimenti, Lara Simone Suárez, Lisa Traynor, Association for Women in Mathematics Series, 27, Research Directions in Symplectic and Contact Geometry and Topology, 2021, 245
Justin Murray, Dan Rutherford, “Legendrian DGA Representations and the Colored Kauffman Polynomial”, SIGMA, 16 (2020), 017, 33 pp.
Maxim Kazarian, Ricardo Uribe-Vargas, “Characteristic points, fundamental cubic form and Euler characteristic of projective surfaces”, Mosc. Math. J., 20:3 (2020), 511–530
Limouzineau M., “On Legendrian Cobordisms and Generating Functions”, J. Knot Theory Ramifications, 29:3 (2020), 2050008
Leverson C., Rutherford D., “Satellite Ruling Polynomials, Dga Representations, and the Colored Homfly-Pt Polynomial”, Quantum Topol., 11:1 (2020), 55–118