Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures
Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures
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传奇链接前沿的组合学和 Arnold 4 猜想
DOI：
10.1070/rm2005v060n01abeh000808
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
P. Pushkar
中科院分区：
文献类型：
--
作者：
Yu. V. Chekanov;P. Pushkar
平面上的每条凸光滑曲线至少有四个点，在这些点上曲线的曲率具有局部极值。如果曲线是通用曲线，则它具有至少有四个尖点的等距曲线。使用接触拓扑语言，V.I.阿诺德提出了将这些经典结果推广到平面上同向前沿的猜想，即四顶点猜想和四尖点猜想。本文对这些猜想和相关结果进行了证明。除了对 Sturm-Hurwitz 理论的简单推广外，证明的主要成分是本文构建的伪对合理论。该理论描述了圆柱体上锋面的组合结构。还讨论了单参数族中莫尔斯复形的赝对合理论与分岔之间的关系。
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.