Topics in Algebraic Topology Based on Knots

9808955Przytycki这个项目涉及了11年来基于节点(或者更一般地基于嵌入对象的位置)构建止痛拓扑的计划。也就是说，基本构建块被考虑到环境同伦(而不是同伦或同调)。例如，从三维流形中的纽结开始，从四维流形中的曲面开始，等等。这是一个影响深远的程序。到目前为止，人们一直被限制在3-流形上，只对4-流形一瞥。该理论的主要对象是Skein模。最简单的Skein模之一是流形的第一同调群的量子化。一般说来，3-流形的skein模是由适当选择的局部(Skein)关系构成的环的拓扑类上的自由模的商。这种情况有点像100年前(在1895年Poincare的基本论文《分析》之前)的代数拓扑学。甚至在三年前，人们只能计算几个孤立的例子，希望这个理论在未来能上升为一个美丽而强大的理论(唯一的例外是在曲面和区间的乘积的第三个弯模上构造Hopf代数的Turaev-Przytycki)。在过去的三年里，这种情况已经开始发生变化，大多数的进展涉及Kauffman括号斜线模及其与三维流形的基本群和流形上的双曲结构的特征簇之间的关系。正如莱布尼茨在1679年预见的那样，拓扑学是一门分析几何图形的艺术，它只考虑几何图形的位置，而不考虑大小(莱布尼茨将这种艺术称为“几何图形”)。在更现代的语言中，拓扑学描述的是伸展和弯曲的空间(但不允许剪切和粘贴)。由于它们的灵活性，拓扑空间很难分析。在1895年的基本论文《分析西图斯》中，亨利·庞加莱将拓扑空间与在空间变形下不变的代数对象联系在一起(他称他的对象为同调和同伦群)。代数拓扑学起源于庞加莱的工作。纽结理论是拓扑学中最古老的分支，由A·范德蒙于1771年首次提出。它研究圆(比如一根末端粘在一起的绳子)在空间中的位置。纽结理论也植根于物理学。W·汤姆森(开尔文勋爵)在1867年提出了涡旋原子理论：原子是打结的乙醚管。他的朋友P.G.Tait的目的是描述粒子的物理和化学性质，以及相关结的性质。虽然旋涡理论很快就被摒弃了，但纽结理论很快发展成为拓扑学的一个独立分支。到了20世纪70年代的S，一些思想结点理论已经过时了。因此，当1984年沃恩·琼斯发现了纽结的新的代数不变量(即琼斯多项式)时，人们感到惊讶。琼斯的工作是一项突破，为古老的猜想提供了解决方案。在与莱布尼茨相同的精神下，人们会将植根于琼斯的构造和德因费尔德关于量子群的工作的数学分支称为“代数场所”。它涵盖了纽结和3-流形的量子不变量理论，基于纽结的代数拓扑，q-变形，量子群，以及与代数几何、非对易几何和统计力学的重叠。

9808955Przytycki This project concerns the 11-year-old program of building analgebraic topology based on knots (or more generally on the position ofembedded objects). That is, the basic building blocks are consideredup to ambient isotopy (not homotopy or homology). For example, onestarts from knots in 3-manifolds, surfaces in 4-manifolds, etc. Thisis a far-reaching program. Until now, one has been limited to 3-manifolds,with only a glance towards 4-manifolds. The principal objects of thetheory are skein modules. One of the simplest skein modules is aquantization of the first homology group of a manifold. In general,skein modules of 3-manifolds are quotients of free modules over ambientisotopy classes of links by properly chosen local (skein) relations.The situation is somewhat reminiscent of that of ``classical'' algebraictopology 100 years ago (before Poincare's fundamental paper ``AnalysisSitus,'' in 1895. Even three years ago one was able to compute only afew isolated examples, with the hope that the theory would rise in thefuture to a beautiful and powerful theory (the only exception was theTuraev-Przytycki construction of the Hopf algebra on the third skeinmodule of the product of a surface and the interval). The situationhas started to change in the last three years, and most of the progressconcerns the Kauffman bracket skein module and its relation to charactervarieties of the fundamental group of a 3-manifold and the hyperbolicstructure on a manifold. Topology, as foreseen by Leibniz in 1679, is the art of analyzinggeometrical figures taking into account their position only; it does nottake magnitudes into consideration (Leibniz called this art ``geometrysitus''). In more modern language, topology describes spaces up tostretching and bending (but cutting and pasting is not allowed). Becauseof their flexibility, topological spaces are hard to analyze. In hisfundamental paper of 1895, Analysis Situs, Henri Poincare associatedtopological spaces with algebraic objects that are invariant under spacedeformation (he called his objects homology and homotopy groups). Thefield of algebraic topology arose from the work of Poincare. Knot theoryis the oldest branch of topology, first considered by A. Vandermonde in1771. It studies the position of a circle (say a piece of rope with endsglued together) in space. Knot theory also has its roots in physics.W. Thomson (Lord Kelvin) proposed, in 1867, a theory of vortex atoms:that atoms were knotted tubes of ether. It was an aim of his friend P.G.Tait to describe the physical and chemical properties of particles interms of the properties of related knots. Although the vortex theorywas soon rejected, knot theory quickly developed to become an independentbranch of topology. By the 1970's, some thought knot theory was out otsteam. Thus it came as a surprise when in 1984 Vaughan Jones discoverednew algebraic invariants of knots (i.e., Jones polynomials). Jones'work was a breakthrough, providing solutions to old conjectures. In thesame spirit as Leibniz, one would call the branch of mathematics that hasits roots in Jones' construction, and Drinfeld's work on quantum groups,``algebra situs.'' It encompasses the theory of quantum invariants ofknots and 3-manifolds, algebraic topology based on knots, q-deformations,quantum groups, and overlaps with algebraic geometry, non-commutativegeometry, and statistical mechanics.***