Algebraic Knots and Representation Theory

拓扑学中一个长期存在的问题是通过询问一个给定的结离解结（即可以被拉开看起来像一个普通的圆）有多远来对结（由绳子在空间中缠绕并自我闭合形成的闭环）进行分类。这个问题（结论主题的基础）在物理学（量子理论）、化学（分子结）和生物学（DNA结）中都有影响。对结进行分类的一个中心工具，实际上在许多拓扑问题中，是给一个结分配一个不变量：如果两个结的不变量不同，那么它们就是不同的。目标是找到能够区分不同结点的鲁棒不变量。本项目探索了新型的结不变量，并延续了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、组合学和表示理论等数学领域的技术。该项目的重点是由平面上的代数曲线与以曲线奇点为中心的小球体相交而产生的结和链接类（这种结和链接称为代数）。在这个项目中，PI将研究代数结点和链路的拓扑不变量与与相应曲线相关的某些代数和组合对象之间的相互作用。量子结不变量已被证明是低维拓扑中的一个强大工具。对于每个结，可以将多项式与一个变量（如Alexander多项式或Jones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Cherednik代数的某些表示，并证明了一些分级子空间的维数与HOMFLY系数相匹配。Khovanov和Rozansky引入了另一种称为HOMFLY同调的向量空间集合，使得HOMFLY系数以其维数交替和的形式表示。这两种构造的相似性表明，对于一个环面结，Khovanov-Rozansky同构可能同构于一个带有额外分级（或过滤）的有理Cherednik代数的表示。这个猜想已经在许多例子中得到了证实，但总的来说仍然是开放的。PI计划使用有理Cherednik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路。其他结同源理论，如heegaard - flower同源，也将被研究。

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.