Hard Unknots and Collapsing Tangles
Hard Unknots and Collapsing Tangles
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硬结和塌陷的缠结
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
S. Lambropoulou
中科院分区：
文献类型：
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作者：
L. Kauffman;S. Lambropoulou
本文给出了无穷多个难解的非结图的例子，在这个意义上说，在简化之前，这些图需要用Reidemister动作来使之更复杂。为了构造这些图，我们证明了刻画两个有理缠结和的分子为无结的定理。关键定理表明，两个有理缠结之和的分子[P/Q]和[R/S]是解结的当且仅当PS+QR的绝对值等于1。本文将这些结果用于研究过程DNA重组，寻找最小尺寸的结图，推广到折叠到结和解结，以及寻找任意高复杂性的解结。这篇论文是自成体系的，回顾了有理纠缠理论，最后一节讨论了论文主题与拓扑学和数论的其他方面的关系。当前版本的论文包含对先前(和出版的)版本的图22的更正，对关于Farey分数的部分的小扩展，以及对Henrich和Kauffman的论文《解开结》的更新参考。
This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is an unknot. The key theorem shows that the numerator of the sum of two rational tangles [P/Q] and [R/S] is unknotted if and only if PS + QR has absolute value equal to 1. The paper uses these results in studying processive DNA recombination, finding minimal size unknot diagrams, generalizing to collapses to knots as well as to unknots, and in finding unknots with arbirarily high complexity. The paper is self-contained, with a review of the theory of rational tangles and a last section on relationships of the theme of the paper with other aspects of topology and number theory. The present version of the paper contains a correction to Figure 22 of the previous (and published) version, a small expansion to the section on Farey fractions, and an updated reference to the paper "Unknotting unknots" by Henrich and Kauffman.