Topics in Algebraic Topology Based on Knots

小行星9808955 这个项目关注的是11年前的基于节点（或更一般地基于嵌入对象的位置）构建嵌入式拓扑的程序。 也就是说，基本的构建块被认为是环境合痕（不是同伦或同调）。 例如，从3-流形中的节点，4-流形中的曲面等开始，这是一个意义深远的程序。 到目前为止，人们一直局限于3-流形，只有一眼对4-流形。 该理论的主要对象是扭模。 最简单的绞模之一是流形的第一同调群的量子化。 一般来说，3-流形的绞链模是自由模在环拓扑类上通过适当选择的局部（绞链）关系的等价物。这种情况有点让人想起100年前的“经典”代数拓扑（在1895年庞加莱的基本论文“AnalysisSitus”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成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skein模上构造Hopf代数的Turaev-Przytycki）。 近三年来，这种情况开始发生变化，主要的进展是关于Kauffman括号skein模及其与3-流形基本群的特征标簇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，是一种分析几何图形的艺术，只考虑它们的位置;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，拓扑学描述了拉伸和弯曲的空间（但不允许剪切和粘贴）。 由于拓扑空间的灵活性，很难分析。 在1895年的基础论文《分析Situs》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起来（他称他的对象为同调和同伦群）。 领域的代数拓扑产生的工作庞加莱。 纽结理论是拓扑学中最古老的分支，最早由A.范德蒙德于1771年。 它研究了一个圆（比如一根两端粘在一起的绳子）在空间中的位置。 纽结理论也有它的物理根源。汤姆逊（开尔文勋爵）在1867年提出了涡旋原子理论：原子是以太的打结管。 这是一个目的，他的朋友P.G.泰特描述的物理和化学性质的粒子interms的性质有关的结。 虽然旋涡理论很快被拒绝，纽结理论很快发展成为一个独立的分支拓扑学。 到了20世纪70年代，有些人认为纽结理论已经过时了。 因此，令人惊讶的是，1984年沃恩·琼斯发现了新的纽结代数不变量（即，Jones多项式）。 琼斯的工作是一个突破，为旧的难题提供了解决方案。 本着与莱布尼茨相同的精神，人们会把数学的分支称为“代数位置”，它的根源在于琼斯的构造和德林费尔德关于量子群的工作。它包含了纽结和三维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，非交换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.***