Prime knots and tangles
Prime knots and tangles
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主要结和缠结
DOI：
10.1090/s0002-9947-1981-0621991-2
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发表时间：
1981
影响因子：
1.3
通讯作者：
W. Lickorish
中科院分区：
文献类型：
--
作者：
W. Lickorish
研究了证明经典纽结或环是素的一种方法。该方法包括一起识别两个主要缠结的边界。探讨了素缠结的构造方法和实例。导论.本文探讨了[KL]中简要介绍的素缠结的概念。这里表明，将两个素数缠结相加总是产生一个素数结或链。这里一个缠结是两个弧跨越一个3球，这样的缠结是素数，如果它不包含打结的球对，它的弧不能被一个圆盘分开。给出了一些素数缠结的原型例子（图2），以及使用它们创建无限多个素数缠结的方法（定理3）。然后，使用素数缠结的想法成为一个强大的机器，很好地补充其他方法，在生产素数结和链接。最后证明了（定理5）素缠结的概念在双分支覆盖中有一个非常自然的解释。该文件应被解释为在P.L.或平滑类别。除了关于双分支覆盖的部分之外，所有使用的方法都是三维流形初等理论的直接（最内层圆盘）技术。没有一个证明是困难的;这篇论文的目的是为了证明重要性而不是复杂性。事实上，为了避免不必要的复杂化，素缠结的各种可能的推广还没有被开发出来（例如n个弧，n > 2，或者带孔的球中的弧在四个点上满足每个边界分量）。作者希望记录下他对加州大学圣巴巴拉分校的感激之情，感谢他们为我提供了写作本书的机会、设施和灵感。
A study is made of a method of proving that a classical knot or link is prime. The method consists of identifying together the boundaries of two prime tangles. Examples and ways of constructing prime tangles are explored. Introduction. This paper explores the idea of the prime tangle that was briefly introduced in [KL]. It is here shown that summing together two prime tangles always produces a prime knot or link. Here a tangle is just two arcs spanning a 3-ball, and such a tangle is prime if it contains no knotted ball pair and its arcs cannot be separated by a disc. A few prototype examples of prime tangles are given (Figure 2), together with ways of using them to create infinitely many more (Theorem 3). Then, usage of the prime tangle idea becomes a powerful machine, nicely complementing other methods, in the production of prime knots and links. Finally it is shown (Theorem 5) that the idea of the prime tangle has a very natural interpretation in terms of double branched covers. The paper should be interpreted as being in either the P.L. or smooth category. With the exception of the section on double branched covers, all the methods used are the straightforward (innermost disc) techniques of the elementary theory of 3-manifolds. None of the proofs is difficult; the paper aims for significance rather than sophistication. Indeed various possible generalizations of the prime tangle have not been developed (e.g. n arcs, for n > 2, or arcs in a ball-with-holes meeting each boundary component in four points) in order to avoid unnecessary complication. The author wishes to record his gratitude to the University of California at Santa Barbara for providing opportunity, facilities and inspiration for the writing of this