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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 1(361), Pages 99–154 (Mi umn1390)  

This article is cited in 38 scientific papers (total in 38 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

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.

DOI: https://doi.org/10.4213/rm1390

Full text: PDF file (827 kB)
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English version:
Russian Mathematical Surveys, 2005, 60:1, 95–149

Bibliographic databases:

UDC: 514.7+515.16
MSC: Primary 57M25, 57R70, 34C23; Secondary 37G10, 57R17, 58K05, 58K10, 57M27, 53D10, 14H50
Received: 20.05.2004

Citation: P. E. Pushkar', Yu. V. Chekanov, “Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures”, Uspekhi Mat. Nauk, 60:1(361) (2005), 99–154; Russian Math. Surveys, 60:1 (2005), 95–149

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    This publication is cited in the following articles:
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    14. Sandon Sh., “Contact Homology, Capacity and Non-Squeezing in $\mathbb{R}^{2n}\times S^1$ via Generating Functions”, Ann Inst Fourier (Grenoble), 61:1 (2011), 145–185  crossref  mathscinet  zmath  isi  scopus
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    19. Lenhard Ng, Daniel Rutherford, “Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra”, Algebr. Geom. Topol, 13:5 (2013), 3047  crossref  mathscinet  zmath  isi  scopus
    20. J.M. Sabloff, Lisa Traynor, “Obstructions to Lagrangian cobordisms between Legendrians via generating families”, Algebr. Geom. Topol, 13:5 (2013), 2733  crossref  mathscinet  zmath  isi  scopus
    21. Lavrov M., Rutherford D., “On the S-1 X S-2 Homfly-Pt Invariant and Legendrian Links”, J. Knot Theory Ramifications, 22:8 (2013), 1350040  crossref  mathscinet  zmath  isi  scopus
    22. Henry M.B., Rutherford D., “A Combinatorial Dga for Legendrian Knots From Generating Families”, Commun. Contemp. Math., 15:2 (2013), 1250059  crossref  mathscinet  zmath  isi  elib  scopus
    23. Chongchitmate W., Ng L., “An Atlas of Legendrian Knots”, Exp. Math., 22:1 (2013), 26–37  crossref  mathscinet  zmath  isi  elib  scopus
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    25. Casey E.E., Henry M.B., “Computing Homology Invariants of Legendrian Knots”, J. Knot Theory Ramifications, 23:11 (2014), 1450056  crossref  mathscinet  zmath  isi  scopus
    26. J.A.. Thornby, R.S.. MacKay, M.A.. Williams, “Mathematical principles for the design of isostatic mount systems for dynamic structures”, IMA J Appl Math, 2015, hxv020  crossref  mathscinet  isi  scopus
    27. Henry M.B., Rutherford D., “Ruling Polynomials and Augmentations Over Finite Fields”, 8, no. 1, 2015, 1–37  crossref  mathscinet  zmath  isi  scopus
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    29. Henry M.B., Rutherford D., “Equivalence classes of augmentations and Morse complex sequences of Legendrian knots”, Algebr. Geom. Topol., 15:6 (2015), 3323–3353  crossref  mathscinet  zmath  isi  scopus
    30. Cornwell Ch., Ng L., Sivek S., “Obstructions to Lagrangian concordance”, Algebr. Geom. Topol., 16:2 (2016), 797–824  crossref  mathscinet  zmath  isi  elib  scopus
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    34. I. A. Dynnikov, M. V. Prasolov, “Rectangular diagrams of surfaces: representability”, Sb. Math., 208:6 (2017), 791–841  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    35. Ng L., Rutherford D., Shende V., Sivek S., “The Cardinality of the Augmentation Category of a Legendrian Link”, Math. Res. Lett., 24:6 (2017), 1845–1874  crossref  mathscinet  zmath  isi
    36. Leverson C., “Augmentations and Rulings of Legendrian Links in #(K) (S-1 X S-2)”, Pac. J. Math., 288:2 (2017), 381–423  crossref  mathscinet  zmath  isi
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    38. 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
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