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Russian Mathematical Surveys, 2005, Volume 60, Issue 1, Pages 95–149
DOI: https://doi.org/10.1070/RM2005v060n01ABEH000808
(Mi rm1390)
 

This article is cited in 58 scientific papers (total in 58 papers)

Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures

P. E. Pushkar'a, Yu. V. Chekanovb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Moscow Center for Continuous Mathematical Education
References:
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.
Received: 20.05.2004
Bibliographic databases:
Document Type: Article
UDC: 514.7+515.16
MSC: Primary 57M25, 57R70, 34C23; Secondary 37G10, 57R17, 58K05, 58K10, 57M27, 53D10, 14H50
Language: English
Original paper language: Russian
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
Citation in format AMSBIB
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\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
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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:
    1. 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  crossref
    2. Wenyuan Li, “Existence of generating families on Lagrangian cobordisms”, Math. Ann., 2024  crossref
    3. Ivan Dynnikov, Vladimir Shastin, “Distinguishing Legendrian knots with trivial orientation-preserving symmetry group”, Algebr. Geom. Topol., 23:4 (2023), 1849–1889  mathnet  crossref
    4. Vladimir Chernov, Rustam Sadykov, “Conjectures about virtual Legendrian knots and links”, Journal of Geometry and Physics, 193 (2023), 104962  crossref
    5. 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  crossref
    6. Petya Pushkar, Misha Temkin, “Enhanced Bruhat Decomposition and Morse Theory”, International Mathematics Research Notices, 2023:19 (2023), 16837  crossref
    7. Dylan Cant, “Remarks on the oscillation energy of Legendrian isotopies”, Geom Dedicata, 217:5 (2023)  crossref
    8. Dynnikov I., Prasolov M., “Rectangular Diagrams of Surfaces: Distinguishing Legendrian Knots”, J. Topol., 14:3 (2021), 701–860  crossref  mathscinet  isi
    9. Haiden F., “Flags and Tangles”, Quantum Topol., 12:3 (2021), 461–505  crossref  mathscinet  isi
    10. Le Roux F., Seyfaddini S., Viterbo C., “Barcodes and Area-Preserving Homeomorphisms”, Geom. Topol., 25:6 (2021), 2713–2825  crossref  mathscinet  isi
    11. Haiden F., “Legendrian Skein Algebras and Hall Algebras”, Math. Ann., 381:1-2 (2021), 631–684  crossref  mathscinet  isi
    12. 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  crossref
    13. Justin Murray, Dan Rutherford, “Legendrian DGA Representations and the Colored Kauffman Polynomial”, SIGMA, 16 (2020), 017, 33 pp.  mathnet  crossref
    14. Maxim Kazarian, Ricardo Uribe-Vargas, “Characteristic points, fundamental cubic form and Euler characteristic of projective surfaces”, Mosc. Math. J., 20:3 (2020), 511–530  mathnet  crossref
    15. Limouzineau M., “On Legendrian Cobordisms and Generating Functions”, J. Knot Theory Ramifications, 29:3 (2020), 2050008  crossref  mathscinet  isi
    16. Leverson C., Rutherford D., “Satellite Ruling Polynomials, Dga Representations, and the Colored Homfly-Pt Polynomial”, Quantum Topol., 11:1 (2020), 55–118  crossref  mathscinet  isi
    17. Lipman E.R., Sabloff J.M., “Lagrangian Fillings of Legendrian 4-Plat Knots”, Geod. Dedic., 198:1 (2019), 35–55  crossref  mathscinet  zmath  isi  scopus
    18. Uribe-Vargas R., “On Projective Umbilics: a Geometric Invariant and An Index”, J. Singul., 17 (2018), 81–90  crossref  mathscinet  zmath  isi  scopus
    19. Rutherford D., Sullivan M.G., “Generating Families and Augmentations For Legendrian Surfaces”, Algebr. Geom. Topol., 18:3 (2018), 1675–1731  crossref  mathscinet  zmath  isi  scopus
    20. Shende V. Treumann D. Zaslow E., “Legendrian knots and constructible sheaves”, Invent. Math., 207:3 (2017), 1031–1133  crossref  mathscinet  zmath  isi  scopus
    Citing articles in Google Scholar: Russian citations, English citations
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