Non-trivial positive braids have positive signature
Non-trivial positive braids have positive signature
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不平凡的积极辫子有积极的签名
DOI：
10.1016/0040-9383(82)90014-3
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发表时间：
1982
期刊：
影响因子：
--
通讯作者：
L. Rudolph
中科院分区：
文献类型：
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作者：
L. Rudolph
N-弦辫子群B中的一个(严格的)正辫子，是可以写成p=i‘的辫子。Lu，，，~，作为一个辫子，它不使用B.i=L的生成元o&的逆(并且确实使用每个ok)。(严格)正辫子的闭包p^是一个有向环，称为(严格)正闭辫子。这类链环有许多有趣的性质：例如(1)严格正闭辫子是纤维链环(特别是作为纽结的正闭辫子是纤维结环)[L，31；(2)它们都出现在C2中合适的3-球面与复代数曲线的交点上[21；(3)它们包括由Bman和Williams[L]研究的Lorenz链环，特别是表现为复代数曲线奇点环的迭代环面链环。正是在我研究(2)的过程中，我有机会问，是否可以证明关于Coobordism类的纽结是正闭的辫子。在与琼·比尔曼和罗伯特·威廉姆斯交谈时，我提出，我或许能够证明，洛伦兹结即使不是微不足道的，也不是薄片。这是真的：随后不久，本说明标题中断言的更一般命题的证据就出来了。我要感谢比尔曼教授和威廉姆斯教授的鼓励。
A (strictly) positive braid in the n-string braid group B, is one which can be written p= I’! lu,,,~, as a braid word which doesn’t use the inverses of the generators o& of B. i= l (and which does use each ok). The closure p^ of a (strictly) positive braid is an oriented link called a (strictly) positive closed braid. Such links have many interesting properties: eg (1) strictly positive closed braids are fibred links (in particular, a positive closed braid which is a knot is a fibred knot)[l, 31;(2) they all occur as intersections of suitable 3-spheres in C2 with complex algebraic curves [21;(3) they include among them the Lorenz links investigated by Birman and Williams [l], in particular, the iterated torus links which appear as links of singular points of complex algebraic curves. It was in the course of my investigation into (2) that I had occasion to ask if anything could be proved about the cobordism classes of knots which are positive closed braids. In talking with Joan Birman and Robert Williams, I proposed that I might be able to show that a Lorenz knot, if not trivial, was not slice. That is true: and a proof of the more general proposition asserted in the title of this note followed shortly. I wish to thank Professors Birman and Williams for their encouragement.