This article is cited in 58 papers

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

P. E. Pushkar'^a, Yu. V. Chekanov<sup>b

a</sup> 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.

UDC: 514.7+515.16

MSC: Primary 57M25, 57R70, 34C23; Secondary 37G10, 57R17, 58K05, 58K10, 57M27, 53D10, 14H50

Received: 20.05.2004

DOI: 10.4213/rm1390

 English version: Russian Mathematical Surveys, 2005, 60:1, 95–149

Bibliographic databases:

• 
• 
• 
• 
• 
• 
•