Classical Topology Inside Quantum Invariants

量子不变量中的经典拓扑

• 批准号: 0072857
• 负责人: Lev Rozansky
• 金额: $ 6.53万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2001-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072857&HistoricalAwards=false
• 关键词: Classical; Topology; Inside; Quantum; Invariants

Proposal: 0072857An important problem of the 3d topology is how to establish a relation between the old "classical" invariants of 3-manifold and knots such as the Casson-Walker invariant and the Alexander polynomial, and the new "quantum" invariants such as the Reshetikhin-Turaev invariant and the Jones polynomial. The existing evidence suggests that quantum invariants can be disassembled into simpler pieces, the first of which are known classical invariants, whereas the others are their close cousins which may also have a nice direct interpretation in the framework of classical topology. The invariants that appear in the decomposition of quantum invariants, are the so-called Vassiliev or finite type invariants. Therefore it turns out that Vassiliev invariants are a link between the quantum and classical invariants. I propose to study the properties of Vassiliev invariants appearing in the decomposition of quantum invariants and especially their relation to classical topology. In particular, I propose to study the new polynomial Vassiliev invariants that are hidden inside the colored Jones polynomials. These new polynomial invariants of knots and links seem to be the descendants of the Alexander polynomial and therefore, similarly to their ancestor, they may have a direct topological interpretation. The first direct descendant of the Alexander polynomial is a 2-variable polynomial with integer coefficients. Path integral arguments suggest that it is a knot analog of the Casson-Walker invariant of rational homology spheres.The topological classification of 3-manifolds and knots is still an open problem. A natural way to approach it is to construct as many invariants of knots and 3-manifolds as one would need in order to distinguish between any non-equivalent objects. In recent years we witnessed an outburst of new "quantum" invariants (Jones polynomial and Reshetikhin-Turaev invariant in particular). These invariants are more powerful than the old Alexander polynomial, yet they are relatively easy to compute. However, their application towards the solution of topological problems is impeded by the fact that their definition is purely combinatorial and has no obvious connection to the well-established methods of classical topology. Existing evidence suggests that quantum invariants are packed up with an infinite number of simpler invariants the first of which are well-known classical invariants. Therefore a study of quantum invariants (especially in the semi-classical approximation when they unpack) may produce an infinite number of new classical invariants which may be enough to distinguish all 3-manifolds and knots. I propose to study the properties of new simple invariants that come from the quantum ones. In particular, I will study the decomposition of the colored Jones polynomial into an infinite sequence of simpler polynomials which seem to be the descendants of the Alexander polynomial.

提案：0072857AN 3D拓扑的重要问题是如何在3-manifold的旧“古典”不变性与卡森 - 沃克不变的结与亚历山大多项式和新的“量子”不变性之间建立关系，例如Reshetikhin-turaev Invaraiant and the varaimiant and the varaimiant and jose novariant and the jones novariant and the jones。现有的证据表明，量子不变性可以分解为更简单的作品，其中第一个是已知的古典不变式，而其他的是它们的亲密表亲，在古典拓扑框架中也可能有一个不错的直接解释。 出现在量子不变的分解中的不变性是所谓的Vassiliev或有限类型不变的。因此，事实证明，Vassiliev不变性是量子和经典不变性之间的联系。 我建议研究出现在量子不变的分解中，尤其是它们与古典拓扑的关系中出现的Vassiliev不变性的特性。特别是，我建议研究隐藏在彩色琼斯多项式中的新多项式vassiliev。这些新的结和链接的多项式不变性似乎是亚历山大多项式的后代，因此，与他们的祖先类似，它们可能具有直接的拓扑解释。亚历山大多项式的第一个直接后代是带有整数系数的2个多项式。路径的整体论点表明，它是Casson-Walker不变的理性同源性领域的结节。三键和结的拓扑分类仍然是一个空旷的问题。一种自然的方法是，为了区分任何非等效对象，构建与人所需的许多不变的结和3个manifolds一样。近年来，我们目睹了新的“量子”不变的爆发（尤其是琼斯多项式和Reshetikhin-turaev不变）。这些不变的人比旧的亚历山大多项式更强大，但是它们相对容易计算。但是，它们在解决拓扑问题方面的应用受到了以下事实的影响：它们的定义纯粹是组合，并且与公认的古典拓扑方法没有明显的联系。 现有的证据表明，量子不变式被无限数量的简单不变式包装，其中第一个是众所周知的古典不变式。因此，对量子不变性的研究（尤其是在拆开包装时在半古典近似中）可能会产生无限数量的新经典不变性物质，这可能足以区分所有3个序列和结。我建议研究来自量子的新简单不变的属性。特别是，我将研究有色琼斯的多项式分解为简单多项式的无限序列，这些序列似乎是亚历山大多项式的后代。