Algebraic Knot Homology

利用代数几何构造结不变量。回想一下，对于每一对李代数和一个表示，我们都可以关联一个结点的Reshetikhin-Turaev不变量。例如，琼斯多项式就是这样一个不变量。在一般线性群和标准表示的情况下，PI和Joel Kamnitzer对这些不变量进行了分类，这意味着他们为每个结分配了一个（双阶）同调群，其欧拉特征是原始不变量。这类似于拓扑空间的（奇异）同调“分类”欧拉特征的方式。它们的构建涉及对某些旗状品种上相干束的（衍生）范畴的研究。所有这些都与Khovanov和Rozansky早期的开创性工作相似，他们使用代数和组合结构发现了这种分类。利用代数几何方法，研究者和他的同事们对如何分类一些剩余的Reshetikhin-Turaev不变量有一个猜想。例如，检查Reidemeister步骤2下的不变性，需要证明（派生的）范畴之间的某些积分变换是等价的。这种等价的存在，包括塞德尔-托马斯球面扭曲，是一个活跃的研究领域，本身就很有趣。打结最简单的例子是把鞋带缠在一起，然后把两端粘在一起。低维拓扑中的一个基本问题是确定两个这样的结何时不同（您可以在不切断它们的情况下移动结）。一种方法是为每个结分配一个数字（一个不变量），这样如果两个结分配了不同的数字，那么它们一定是不同的。如何分配这样的数字是一个深刻的问题，它以令人惊讶的方式与数学的各个领域有关。例如，Reshetikhin-Turaev不变量是从表示理论（这是对矩阵的研究）中得到的。在物理学和弦理论的启发下，也可以用代数和其他方法来构造这种不变量。拟议的研究提出了另一种使用代数几何（即多项式方程系统的研究）的方法。这种方法可以扩展、统一和阐明其他结构。

The proposal is to construct knot invariants using algebraic geometry.Recall that to each pair of a Lie algebra and a representation one can associate a Reshetikhin-Turaev invariant of knots. For example, the Jones polynomial is one such invariant. In the case of the general linear group and the standard representation the PI and Joel Kamnitzer categorify these invariants, meaning that to each knot they assign a (bi-graded) homology group whose Euler characteristic is the original invariant. This is analogous to the way (singular) homology of topological spaces "categorifies" the Euler characteristic. Their construction involves studying the (derived) category of coherent sheaves on certain flag-like varieties. All this parallels the earlier pioneering work of Khovanov and Rozansky who discovered such categorifications using algebraic and combinatorial constructions. Using the algebraic geometric approach the investigator and his colleagues have a conjecture for how to categorify some of the remaining Reshetikhin-Turaev invariants. Checking, for instance, invariance under Reidemeister move 2 involves proving that certain integral transforms between (derived) categories are equivalences.The existence of such equivalences, which include Seidel-Thomas spherical twists, is a lively area of research which is interesting in itself.The simplest example of a knot is obtained by taking a tangled-up shoe lace and glueing the ends together. A fundamental question in low-dimensional topology is to determine when two such knots are different (you are allowed to move the knots around without cutting them). One way to do this is to assign a number (an invariant) to each knot so that if two knots are assigned different numbers then they must be different. How to assign such numbers is a deep problem which is related to various areas of mathematics in surprising ways. For example, the Reshetikhin-Turaev invariants are obtained from representation theory (which is the study of matrices). There are also constructions of such invariants using algebra and others that are inspired by physics and string theory. The proposed research suggests yet another approach using algebraic geometry (which is the study of systems of polynomial equations). This approach may extend, unify and shed more light on the other constructions.