On transversally simple knots
On transversally simple knots
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关于横向简单结
DOI：
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发表时间：
1999
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通讯作者：
Nancy C. Wrinkle
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作者：
J. Birman;Nancy C. Wrinkle
本文研究了IR3中横切于标准接触结构的纽结，将拓扑纽结理论中的技巧引入到它们的横向分类中。我们称一个横截纽结TK是横单的，如果它由它的拓扑纽结类型K和它的Bennequin数决定。主要定理断言，其相关K满足交换可约性条件的任何T_K都是横向简单的。作为应用，我们给出了Eliashberg[El91]的一个定理的新证明，该定理断言纽结是横单的，并且(借助于Menasco[Me99]的一个新定理)推广了Etnyre[Et99]的一个结果，证明了所有迭代环面纽结都是横单的。我们证明了交换可约性的概念是人们可以对K施加的最简单的约束，以证明任何相关的T_K是横向简单的。我们还给出了我们猜想不是横切简单的横截纽结对的例子。
This paper studies knots that are transversal to the standard contact structure in IR 3 , bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type T K is transversally simple if it is determined by its topological knot type K and its Bennequin number. The main theorem asserts that any T K whose associated K satisfies a condition that we call exchange reducibility is transversally simple. As applications, we give a new proof of a theorem of Eliashberg [El91], which asserts that the unknot is transversally simple, and (with the help of a new theorem of Menasco [Me99]) extend a result of Etnyre [Et99] to prove that all iterated torus knots are transversally simple. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on K in order to prove that any associated T K is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.