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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.


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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

Citation in format AMSBIB
\by P.~E.~Pushkar', Yu.~V.~Chekanov
\paper Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures
\jour Uspekhi Mat. Nauk
\yr 2005
\vol 60
\issue 1(361)
\pages 99--154
\jour Russian Math. Surveys
\yr 2005
\vol 60
\issue 1
\pages 95--149

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    1. Melvin P., Shrestha S., “The nonuniqueness of Chekanov polynomials of Legendrian knots”, Geom. Topol., 9 (2005), 1221–1252  crossref  mathscinet  zmath  isi
    2. Lenhard Ng, “A Legendrian Thurston–Bennequin bound from Khovanov homology”, Algebr. Geom. Topol., 5 (2005), 1637–1653  crossref  mathscinet  zmath  isi
    3. Kálmán T., “Braid-positive Legendrian links”, Int. Math. Res. Not., 2006, 14874, 29 pp.  crossref  mathscinet  zmath  scopus
    4. Rutherford D., “The Thurston-Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: The Fuchs conjecture and beyond”, Int. Math. Res. Not., 2006, 78591, 15 pp.  mathscinet  zmath  isi
    5. Ng L.L., Sabloff J.M., “The correspondence between augmentations and rulings for Legendrian knots”, Pacific J. Math., 224:1 (2006), 141–150  crossref  mathscinet  zmath  isi  elib  scopus
    6. Jordan J., Traynor L., “Generating family invariants for Legendrian links of unknots”, Algebr. Geom. Topol., 6 (2006), 895–933  crossref  mathscinet  zmath  isi  scopus
    7. Proc. Steklov Inst. Math., 258 (2007), 178–193  mathnet  crossref  mathscinet  zmath  elib
    8. Kálmán T., “Rulings of Legendrian knots as spanning surfaces”, Pacific J. Math., 237:2 (2008), 287–297  crossref  mathscinet  zmath  isi  scopus
    9. Eliashberg Ya., Fraser M., “Topologically trivial Legendrian knots”, J. Symplectic Geom., 7:2 (2009), 77–127  crossref  mathscinet  zmath  isi  elib  scopus
    10. Ding Fan, Geiges Hansjörg, “The diffeotopy group of $S^1\times S^2$ via contact topology”, Compositio Math., 146:4 (2010), 1096–1112  crossref  mathscinet  zmath  isi  elib  scopus
    11. Civan G., Koprowski P., Etnyre J., Sabloff J.M., Walker A., “Product structures for Legendrian contact homology”, Math. Proc. Camb. Phil. Soc., 150 (2011), 291–311  crossref  mathscinet  zmath  isi  elib  scopus
    12. Henry M.B., “Connections between Floer-type invariants and Morse-type invariants of Legendrian knots”, Pacific J. Math., 249:1 (2011), 77–133  crossref  mathscinet  zmath  isi  elib  scopus
    13. Georgios Dimitroglou Rizell, “Knotted Legendrian surfaces with few Reeb chords”, Algebr. Geom. Topol, 11:5 (2011), 2903  crossref  mathscinet  zmath  isi  scopus
    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
    15. Kalman T., “Inner products on the Hecke algebra of the braid group”, Topology Appl, 158:5 (2011), 643–646  crossref  mathscinet  zmath  isi  elib  scopus
    16. Fuchs D., Rutherford D., “Generating families and Legendrian contact homology in the standard contact space”, Journal of Topology, 4:1 (2011), 190–226  crossref  mathscinet  zmath  isi  scopus
    17. Ng L., “On Arc Index and Maximal Thurston-Bennequin Number”, J Knot Theory Ramifications, 21:4 (2012), 1250031  crossref  mathscinet  zmath  isi  elib  scopus
    18. Mikhail Lavrov, Dan Rutherford, “Generalized normal rulings and invariants of Legendrian solid torus links”, Pacific J. Math, 258:2 (2012), 393  crossref  mathscinet  zmath  isi  scopus
    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
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    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
    24. Etnyre J.B., Ng L.L., Vertesi V., “Legendrian and Transverse Twist Knots”, J. Eur. Math. Soc., 15:3 (2013), 969–995  crossref  mathscinet  zmath  isi  elib  scopus
    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
    28. Bourgeois F., Sabloff J.M., Traynor L., “Lagrangian Cobordisms Via Generating Families: Construction and Geography”, 15, no. 4, 2015, 2439–2477  crossref  mathscinet  zmath  isi  scopus
    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
    31. Leverson C., “Augmentations and rulings of Legendrian knots”, J. Symplectic Geom., 14:4 (2016), 1089–1143  crossref  mathscinet  zmath  isi  scopus
    32. Rizell G.D., “Legendrian ambient surgery and Legendrian contact homology”, J. Symplectic Geom., 14:3 (2016), 811–901  crossref  mathscinet  zmath  isi  elib  scopus
    33. Shende V. Treumann D. Zaslow E., “Legendrian knots and constructible sheaves”, Invent. Math., 207:3 (2017), 1031–1133  crossref  mathscinet  zmath  isi  scopus
    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
    37. Uribe-Vargas R., “On Projective Umbilics: a Geometric Invariant and An Index”, J. Singul., 17 (2018), 81–90  crossref  mathscinet  zmath  isi  scopus
    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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