On the Vassiliev invariants for knots and for pure braids
On the Vassiliev invariants for knots and for pure braids
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关于结和纯辫子的 Vassiliev 不变量
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发表时间：
1997
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通讯作者：
S. Willerton
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作者：
S. Willerton
研究瓦西里耶夫结不变量产生于瓦西里耶夫的工作奇异性理论和微扰陈-西蒙斯理论的维滕。研究瓦西里耶夫不变量的一个原因是，它们给出了看待“量子”结不变量的拓扑方法--即通过推广琼斯多项式而产生的不变量。这篇论文包含了各种结果瓦西里耶夫不变量：贯穿其中的共同主题包括他们的多项式性质，他们的功能，并使用高斯图。第一章考察了Vassiliev不变量的函子性，并描述了它们如何在不同类型的棘手对象上定义，如节点，框架节点和辫子，以及代数结构如何自然产生。给出了有框架纽结理论与无框架纽结理论之间关系的一种显式形式。第2章考虑了一个重要的问题，是否可以天真地从一个组合对象称为重量系统的瓦西里耶夫不变量。一个部分的答案，这是通过显示如何“一半”的步骤，在这样的过渡可以执行规范和明确的。在第3章中，检查了在多达12个交叉的素结上评估的结的前两个非平凡不变量，并通过绘制它们获得了一些令人惊讶的图形。证明了环面纽结的一些结果，将解结数和交叉数与前两个Vassiliev不变量联系起来。论文的后半部分主要研究纯辫子的Vassiliev不变量及其与de Rham同伦理论的联系。第四章给出了一个简单的推导，说明了Vassiliev不变量与纯辫群的下中心列之间的关系。这是用来获得封闭公式的实际数量的不变量的每一个类型。第5章是一个离题的德拉姆同伦理论，并解释了几何之间的联系陈的迭代积分，高阶阿尔巴尼亚流形，沙利文的1-极小模型。本文给出了Chen从1-极小模型求基本群元素积分不变量的方法，并在第六章中用它求低阶纯辫子的Vassiliev不变量，这推广了M. A.伯杰最后，采用类似的方法使用电流，得到了一个组合公式的第二型不变量，这是独立的绕组数。
The study of Vassiliev knot invariants arose from Vassiliev’s work on singularity theory and from the perturbative Chern-Simons theory of Witten. One reason for studying Vassiliev invariants is that they give topological ways of looking at “quantum” knot invariants — that is invariants which arise by generalizing the Jones polynomial. This thesis contains various results on Vassiliev invariants: common themes running through include their polynomial nature, their functoriality, and the use of Gaus diagrams. The first chapter examines the functoriality of Vassiliev invariants and describes how they can be defined on different types of knotty objects such as knots, framed knots and braids, and how algebraic structure naturally arises. An explicit form of the relationship between the framed and unframed knot theory is given. Chapter 2 considers the important question of whether a Vassiliev invariant can be naively obtained from a combinatorial object called a weight system. A partial answer to this is given by showing how “half” of the steps in such a transition can be performed canonically and explicitly. In Chapter 3 the first two non-trivial invariants for knots, evaluated on prime knots up to twelve crossing are examined, and some surprising graphs are obtained by plotting them. A number of results for torus knots are proved, relating unknotting number and crossing number to the first two Vassiliev invariants. The second half of the thesis is concerned primarily with Vassiliev invariants of pure braids and their connection with de Rham homotopy theory. In Chapter 4 a simple derivation is given showing the relationship between Vassiliev invariants and the lower central series of the pure braid groups. This is used to obtain closed formulae for the actual number of invariants of each type. Chapter 5 is a digression on de Rham homotopy theory and explains the geometric connections between Chen’s iterated integrals, higher order Albanese manifolds, and Sullivan’s 1-minimal models. A method of Chen’s for obtaining integral invariants of elements of the fundamental group from a 1-minimal model is given, and in Chapter 6 this is used to find Vassiliev invariants of pure braids at low order: this extends work of M. A. Berger. Finally, a similar method using currents is employed to obtain a combinatorial formula for a type two invariant which is independent of winding numbers.