Algebraic Knot Homology

代数结同调

• 批准号: 0801939
• 负责人: Sabin Cautis
• 金额: $ 10.11万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801939&HistoricalAwards=false
• 关键词: Algebraic; Knot; Homology

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.

该提议是使用代数几何构造结不变量。回想一下，对于每一对李代数和表示，我们可以将结的 Reshetikhin-Turaev 不变量关联起来。例如，琼斯多项式就是这样的不变量之一。在一般线性群和标准表示的情况下，PI 和 Joel Kamnitzer 对这些不变量进行分类，这意味着他们为每个结分配一个（双级）同源群，其欧拉特征是原始不变量。这类似于拓扑空间的（奇异）同源“分类”欧拉特征的方式。它们的构建涉及研究某些旗状品种上连贯滑轮的（派生）类别。所有这些都与霍瓦诺夫和罗赞斯基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。使用代数几何方法，研究者和他的同事对如何对剩余的一些 Reshetikhin-Turaev 不变量进行分类进行了猜想。例如，检查 Reidemeister move 2 下的不变性涉及证明（派生）类别之间的某些积分变换是等价的。这种等价的存在，包括 Seidel-Thomas 球形扭曲，是一个活跃的研究领域，本身就很有趣。最简单的结的例子是通过取一根缠结的鞋带并将两端粘在一起来获得。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（允许您移动结而不切断它们）。一种方法是为每个结分配一个数字（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。如何分配这些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相关。例如，Reshetikhin-Turaev 不变量是从表示论（矩阵研究）获得的。受物理学和弦理论的启发，还可以使用代数和其他方法构造此类不变量。拟议的研究提出了另一种使用代数几何的方法（即多项式方程组的研究）。这种方法可以扩展、统一并进一步阐明其他结构。