Cabling and transverse simplicity
Cabling and transverse simplicity
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布线和横向简单性
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发表时间：
2003
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通讯作者：
K. Honda
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作者：
John B. Etnyre;K. Honda
我们研究绳结类型中的Legendrian结。具体来说，给定一个拓扑结类型/C，我们分析了由K通过布线得到的结类型中的Legendrian结，根据结类型K中的Legendrian结，作为该分析的推论，我们证明了(2,3)-环面结的(2,3)-电缆不是横向简单的，并对该结类型中的横向结进行了分类。这是非横向简单结类型中横向结的第一个分类。我们也对这种结类型的Legendrian结进行了分类，并展示了第一个不失稳的Legendrian结的例子，但它的Thurston-Bennequin不变量在其结类型的Legendrian代表中不是极大的。本文继续用三维接触拓扑方法研究紧密接触3流形中的Legendrian节。在[EH1]中，作者介绍了分析紧密接触3流形中的Legendrian节的一般框架。在那里，我们简化了最初由Eliashberg-Eraser在[EF]中证明的Legendrian解结分类的证明，并给出了Legendrian环面结和图8结的完整分类。在[EH2]中，我们给出了Legendrian结的第一个结构定理，即将Legendrian结的连通和的分析简化为素数和的分析。这产生了大量的非传奇简单结类型。（如果拓扑结类型中的Legendrian结由其Thurston-Bennequin不变量和旋转数决定，则该结类型为Legendrian simple。）此外，我们展示了具有相同Thurston-Bennequin和旋转数的相同拓扑结类型的Legendrian结对，它们在成为Legendrian同位素之前需要任意多次稳定（见[EH2]）。
We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type /C, we analyze the Legendrian knots in knot types ob tained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a non transversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a Legendrian knot that does not destabi lize, yet its Thurston-Bennequin invariant is not maximal among Legendrian representatives in its knot type. In this paper we continue the investigation of Legendrian knots in tight contact 3-manifolds using 3-dimensional contact-topological methods. In [EH1], the authors introduced a general framework for analyzing Legendrian knots in tight contact 3-manifolds. There we streamlined the proof of the classification of Legendrian unknots, originally proved by Eliashberg-Eraser in [EF], and gave a complete classification of Legendrian torus knots and figure eight knots. In [EH2], we gave the first structure theorem for Legendrian knots, namely the reduction of the analysis of connected sums of Legendrian knots to that of the prime summands. This yielded a plethora of non-Legendrian-simple knot types. (A topological knot type is Legendrian simple if Legendrian knots in this knot type are determined by their Thurston-Bennequin invariant and rotation number.) Moreover, we exhibited pairs of Legendrian knots in the same topological knot type with the same Thurston-Bennequin and rotation numbers, which required arbitrarily many stabilizations before they became Legendrian isotopic (see [EH2]).