Classical Topology Inside Quantum Invariants

建议：3D拓扑的一个重要问题是如何在3-流形和节点的旧的“经典”不变量（如Casson-Walker不变量和亚历山大多项式）与新的“量子”不变量（如Reshetikhin-Turaev不变量和Jones多项式）之间建立关系。现有的证据表明，量子不变量可以被分解成更简单的部分，其中第一个是已知的经典不变量，而其他的是它们的近亲，在经典拓扑学的框架中也可能有一个很好的直接解释。 在量子不变量的分解中出现的不变量是所谓的Vassiliev或有限型不变量。因此，瓦西里耶夫不变量是量子和经典不变量之间的联系。 我建议研究性质的瓦西里耶夫不变量出现在量子不变量的分解，特别是它们的关系，经典拓扑。特别是，我建议研究新的多项式Vassiliev不变量是隐藏在有色琼斯多项式。这些新的多项式不变量的纽结和链接似乎是后代的亚历山大多项式，因此，类似于他们的祖先，他们可能有一个直接的拓扑解释。亚历山大多项式的第一直接后代是具有整数系数的二元多项式。路径积分是有理同调球面的Casson-Walker不变量的一个纽结类似物，三维流形和纽结的拓扑分类仍然是一个开放的问题。一个自然的方法是构造尽可能多的纽结和三维流形的不变量，以便区分任何不等价的对象。近年来，我们目睹了新的“量子”不变量（特别是琼斯多项式和Reshetikhin-Turaev不变量）的爆发。这些不变量比旧的亚历山大多项式更强大，但它们相对容易计算。然而，他们的应用对解决拓扑问题的阻碍，他们的定义是纯粹的组合，并没有明显的连接到经典拓扑学的完善的方法。 现有的证据表明，量子不变量是由无数个简单的不变量组成的，其中第一个是众所周知的经典不变量。因此，对量子不变量的研究（特别是在半经典近似下，当它们展开时）可能会产生无限多个新的经典不变量，这些不变量可能足以区分所有的三维流形和纽结。我打算研究来自量子不变量的新的简单不变量的性质。特别是，我将研究的有色琼斯多项式分解成一个无限序列的简单的多项式似乎是后代的亚历山大多项式。

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.