Algebraic Knots and Representation Theory

代数结和表示论

• 批准号: 1559338
• 负责人: Evgeny Gorskiy
• 金额: $ 8.96万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1559338&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.

拓扑学中一个长期存在的问题是通过询问给定的结离解开（即，可以被拉开看起来像普通的圆）有多远来对结（由在空间中缠绕并闭合自身的绳子形成的闭合环）进行分类。 这个问题（纽结理论的基础）在物理学（量子理论）、化学（分子结）和生物学（DNA的打结）中都有意义。 在对纽结进行分类时，以及在许多拓扑问题中，一个核心工具是给一个纽结赋予一个不变量：如果两个纽结的不变量不同，那么它们就不同。 目标是找到能够区分不同节点的鲁棒不变量。 这个项目探索了新类型的结不变量，并继续使用代数工具来定义和研究结不变量的趋势。 PI将使用代数几何，组合数学和表示论等数学领域的技术。该项目的重点是一类结和链接，产生于相交的代数曲线在平面上与一个小球体中心的奇异性曲线（这种结和链接被称为代数）。在这个项目中，PI将研究代数节点和链接的拓扑不变量与与相应曲线相关联的某些代数和组合对象之间的相互作用。量子节点不变量已被证明是低维拓扑学中的一个强大工具。 对于每一个纽结，我们可以将一个多项式与一个变量（如亚历山大多项式或琼斯多项式）或两个变量（如HOMFLY多项式）中的整数系数联系起来。 PI和他的合作者最近发现，对于环面结，HOMFLY多项式中的所有系数实际上都是非负的。 为了证明这一事实，研究了有理数Cherednik代数的某些表示，并证明了某些分次子空间的维数与HOMFLY系数匹配。 Khovanov和Rozansky引入了另一个向量空间集合，称为HOMFLY同调，使得HOMFLY系数表示为它们的维数的交替和。 这两个结构的相似性表明，对于环面结，Khovanov-Rozansky同调可能同构于有理切雷德尼克代数的表示，配备了额外的分级（或过滤）。 这个猜想已经在许多例子中得到了验证，但在一般情况下仍然是开放的。 PI计划使用有理切雷德尼克代数的表示理论来构建环面结的Khovanov-Rozansky同调的显式组合和几何模型，并将其推广到代数结和链接。 其他结同源理论，如Heegaard-Floer同源，也将进行研究。