Topics in Algebraic Topology Based on Knots

这个项目涉及基于结点（或者更一般地基于嵌入对象的位置）构建代数拓扑的11年的程序。也就是说，基本构建块被认为是环境同位素（不是同伦或同源）。例如，从3流形中的结点、4流形中的曲面等入手。这是一个影响深远的计划。到目前为止，人们已经被限制在3流形，只有一瞥到4流形。该理论的主要对象是绞合模块。最简单的串模之一是流形的第一个同调群的量化。一般来说，3流形的绞合模是由适当选择的局部（绞合）关系在连杆的环境拓扑类上的自由模的商。这种情况有点让人想起100年前的“经典”代数拓扑学（在1895年庞加莱发表基础论文《分析现状》之前）。即使在三年前，人们也只能计算出几个孤立的例子，希望这个理论在未来能够成为一个美丽而强大的理论（唯一的例外是在曲面与区间之积的第三个skeinmodule上的Hopf代数的turaev - przytycki构造）。近三年来，这种情况开始发生变化，大多数进展涉及到考夫曼支架串模及其与3流形基本群的特征变异和流形上的双曲结构的关系。拓扑学，正如莱布尼茨在1679年所预见的那样，是一门只考虑几何图形位置的分析艺术；它不考虑大小（莱布尼茨称这种艺术为“几何学”）。在更现代的语言中，拓扑描述空间伸展和弯曲（但剪切和粘贴是不允许的）。由于其灵活性，拓扑空间很难分析。在他1895年的基础论文《分析位置》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（他称他的对象为同调和同伦群）联系起来。代数拓扑学是由庞加莱的工作产生的。结理论是拓扑学中最古老的分支，最早由A. Vandermonde于1771年提出。它研究一个圆（比如一根两端粘在一起的绳子）在空间中的位置。绳结理论在物理学中也有其根源。汤姆逊（开尔文勋爵）在1867年提出了旋涡原子理论：原子是结在一起的乙醚管。他的朋友p.g.泰特的目标是用相关结的性质来描述粒子的物理和化学性质。虽然旋涡理论很快被拒绝，但结理论迅速发展成为拓扑学的一个独立分支。到了20世纪70年代，有些人认为结理论已经过时了。因此，当1984年Vaughan Jones发现结点的新代数不变量（即Jones多项式）时，人们感到很惊讶。琼斯的工作是一个突破，为旧的猜想提供了答案。本着和莱布尼茨一样的精神，人们会把植根于琼斯的构造和德林菲尔德关于量子群的研究的数学分支称为“代数位”。“它涵盖了结点和3流形的量子不变量理论，基于结点的代数拓扑，q -变形，量子群，以及与代数几何，非交换几何学和统计力学的重叠。

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.***