Tait’s conjectures and odd crossing number amphicheiral knots
Tait’s conjectures and odd crossing number amphicheiral knots
复制标题
泰特猜想和奇数交叉数两栖结
DOI：
10.1090/s0273-0979-08-01196-8
复制
发表时间：
2007
影响因子：
1.3
通讯作者：
A. Stoimenow
中科院分区：
文献类型：
--
作者：
A. Stoimenow
我们简要回顾一下泰特猜想的历史。泰特猜想是在120年前他开创性地将最简单的结制成表格的过程中提出的，一个世纪后用琼斯多项式解决了这个猜想。我们再次基于对琼斯多项式的大量研究，用几乎所有奇交叉数的两角结的构造，宣布了一个（可能是他最后遗留的）Tait问题的解决方案。本文还概述了在琼斯多项式非平凡性问题中的应用。1. 结理论起源于19世纪晚期。当时，W.汤姆森（“开尔文勋爵”）、P. G.泰特和J.麦克斯韦传播了漩涡原子理论，试图解释宇宙的结构。他们认为，所有的物质都是由一种超物质以太构成的，而原子是由以太管组成的。打结在此被理解为系一根绳子，然后确定其两端，使系结不能再解开。因此，在构建元素周期表的领域，泰特开始对最简单的结进行编目。他描绘的结（就像我们今天所做的那样）是由一个（光滑的）平面曲线和横向自交或交叉组成的图表。在每一次交叉处，两条线中的一条会越过另一条。术语“最简单”指的是图表的交叉点数量。我们可以将一个结的交叉数定义为其所有图的最小交叉数，并说Tait寻找具有给定（小）交叉数的结的列表。这个列表的目的是用一个图表来表示每个结。这就要求不同图上的结应该是不相等的，也就是说，如果不剪断绳子，就不能把一根（闭合的）绳子从一种方向变成另一种方向。最简单的结如图1所示。最左边的一个，交叉数0，是平凡的结或解结。它有一些特殊的重要性，就像群中的单位元素一样。泰特完成了多达7个过境点的清单。利特尔、柯克曼和后来的康威等人接替了他的工作，继续他的工作。在现代计算机时代，表格已经达到了17个交叉点的结，有数百万个条目，尽管泰特的漩涡原子理论早已被驳回。编者于2007年5月30日收到结表的报告。2000数学学科分类。主要57 m25公路;二级01A55、01A60。
We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last remaining) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An application to the non-triviality problem for the Jones polynomial is also outlined. 1. The first knot tables Knot theory originated in the late 19th century. At that time, W. Thomson (“Lord Kelvin”), P. G. Tait and J. Maxwell propagated the vortex-atom theory in an attempt to explain the structure of the universe. They believed that a supersubstance, ether, makes up all of matter, and atoms are knotted tubes of ether. Knotting is hereby understood as tying a piece of rope and then identifying both its ends so that the tying cannot be any more undone. Thus, in the realm of constructing a periodic table of elements, Tait began the catalogization of the simplest knots. He depicted knots (as we still do today) by diagrams consisting of a (smooth) plane curve with transverse self-intersections, or crossings. At each crossing one of the two strands passes over the other. The term “simplest” refers to the number of crossings of the diagram. We can define the crossing number of a knot as the minimal crossing number of all its diagrams, and say that Tait sought the list of knots with given (small) crossing number. The list meant to present each knot by exactly one diagram. This entails that knots from different diagrams should be inequivalent, in the sense that one cannot turn a (closed) piece of rope knotted one way into one knotted the other way without cutting the rope. The simplest knots are shown in figure 1. The leftmost one, of crossing number 0, is the trivial knot or unknot. It has some special importance, much like the unit element in a group. Tait completed the list up to 7 crossings. Little, Kirkman, later Conway [Co] and others took over and continued his work. In the modern computer age, tables have reached the knots of 17 crossings, with millions of entries, even though Tait’s vortex-atom theory has long been dismissed. An account on knot tabulation is Received by the editors May 30, 2007. 2000 Mathematics Subject Classification. Primary 57M25; Secondary 01A55, 01A60.