Algebraic Knots and Representation Theory

代数结和表示论

• 批准号: 1403560
• 负责人: Evgeny Gorskiy
• 金额: $ 13.46万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1403560&HistoricalAwards=false
• 关键词: Algebraic; Knots; Representation; Theory

A longstanding problem in topology is to classify knots (a closed loop formed from a rope winding in space and closing back on itself) by asking how far a given knot is from being unknotted (that is, can be pulled apart to look like an ordinary circle). This problem (the basis of the subject of knot theory) has implications in physics (quantum theory), chemistry (molecular knots) and biology (knotting of DNA). A central tool in classifying knots, and indeed in many topological questions, is to assign an invariant to a knot: two knots are then different if their invariants are different. The goal is find robust invariants which can distinguish different knots. This project explores new types of knot invariants and continues the trend of using tools from algebra to define and investigate knot invariants. The PI will use techniques from the mathematical fields of algebraic geometry, combinatorics, and representation theory. The focus of the project is on the class of knots and links that arise from intersecting an algebraic curve in the plane with a small sphere centered at the singularity of the curve (such knots and links are called algebraic). In this project the PI will study the interaction between the topological invariants of algebraic knots and links and certain algebraic and combinatorial objects associated to the corresponding curve.Quantum knot invariants have proven to be a powerful tool in low-dimensional topology. To every knot one can associate a polynomial with integer coefficients in one variable (as in the Alexander polynomial or the Jones polynomial) or in two variables (as in the HOMFLY polynomial). It has recently been discovered by the PI and his collaborators that for torus knots all coefficients in the HOMFLY polynomial are in fact nonnegative. To prove this fact, certain representations of the rational Cherednik algebra were studied and it was shown that the dimensions of some graded subspaces match the HOMFLY coefficients. Khovanov and Rozansky introduced another collection of vector spaces, called HOMFLY homology, such that the HOMFLY coefficients are presented as alternating sums of their dimensions. The similarity of the two constructions suggests that for a torus knot Khovanov-Rozansky homology may be isomorphic to a representation of the rational Cherednik algebra, equipped with an extra grading (or filtration). This conjecture has been verified in many examples, but remains open in general. The PI plans to use the representation theory of rational Cherednik algebras for the construction of explicit combinatorial and geometric models for the Khovanov-Rozansky homology of torus knots, and their generalizations to algebraic knots and links. Other knot homology theories, such as Heegaard-Floer homology, will also be studied.

拓扑结构的一个长期存在的问题是，通过询问给定的结距离未结的距离（也就是说，可以拉开一个普通的打结，是由太空中的绳索形成的封闭环（由绳索绕过并自身闭上）来分类圆圈）。 这个问题（结理论主题的基础）对物理学（量子理论），化学（分子结）和生物学（DNA的结）具有影响。 分类结的核心工具，甚至在许多拓扑问题中，是为结分配一个不变性：如果它们的不变性不同，则两个结是不同的。 目标是找到可以区分不同结的强大不变式。 该项目探索了新型的结不变，并继续使用代数的工具来定义和研究结的趋势。 PI将使用代数几何，组合和表示理论的数学领域的技术。该项目的焦点是与曲线奇异性的小球中的代数曲线相交的结和链接（此类结和链接称为代数）。在该项目中，PI将研究代数结与链接的拓扑不变剂以及与相应曲线相关的某些代数和组合对象之间的相互作用。事实证明，Quantum结的不变性已被证明是低维拓扑的强大工具。 每个结都可以将多项式与一个变量（如亚历山大多项式或琼斯多项式）或两个变量（如Homfly多项式中的多项式）相关联。 PI及其合作者最近发现，对于homfly多项式中的所有系数而言，所有系数实际上都是无负的。 为了证明这一事实，研究了理性Cherednik代数的某些表示形式，并显示某些分级子空间的尺寸与Homfly系数匹配。 Khovanov和Rozansky推出了另一个称为Homfly同源性的向量空间集合，因此将Homfly系数作为其尺寸的交替总和表示。 这两个结构的相似性表明，对于圆环结，khovanov-rozansky的同源性可能与配备额外的分级（或过滤）的有理Cherednik代数的表示同构。 该猜想在许多示例中都得到了验证，但总体上仍然是开放的。 PI计划将理性Cherednik代数的表示理论用于构建福音结的Khovanov-Rozansky同源性的显式组合和几何模型，以及它们对代数结和链接的概括。 还将研究其他结的同源理论，例如Heegaard-loer同源性。