Classical Topology Inside Quantum Invariants

三维拓扑学的一个重要问题是如何建立3流形和结点的旧的“经典”不变量（如Casson-Walker不变量和Alexander多项式）与新的“量子”不变量（如Reshetikhin-Turaev不变量和Jones多项式）之间的关系。现有的证据表明，量子不变量可以被分解成更简单的部分，其中第一部分是已知的经典不变量，而其他的是它们的近亲，它们也可能在经典拓扑的框架中有一个很好的直接解释。在量子不变量分解中出现的不变量，是所谓的Vassiliev不变量或有限型不变量。因此，瓦西里耶夫不变量是连接量子不变量和经典不变量的纽带。本文拟研究量子不变量分解中出现的Vassiliev不变量的性质，特别是它们与经典拓扑的关系。特别地，我建议研究隐藏在有色琼斯多项式中的新的多项式Vassiliev不变量。这些结点和连杆的新多项式不变量似乎是亚历山大多项式的后代，因此，与它们的祖先相似，它们可能有直接的拓扑解释。亚历山大多项式的第一个直系后代是一个具有整数系数的2变量多项式。路径积分论证表明它是有理同调球的Casson-Walker不变量的一个结类比。3流形和结点的拓扑分类仍然是一个开放的问题。一种自然的方法是构造尽可能多的结点和3流形的不变量，以便区分任何非等效对象。近年来，我们目睹了新的“量子”不变量（特别是Jones多项式和Reshetikhin-Turaev不变量）的爆发。这些不变量比旧的Alexander多项式更强大，但它们相对容易计算。然而，由于它们的定义是纯组合的，并且与已建立的经典拓扑方法没有明显的联系，因此阻碍了它们在拓扑问题求解中的应用。现有的证据表明，量子不变量是由无数个更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中，当它们解包时）可能会产生无限数量的新的经典不变量，这些不变量可能足以区分所有的3流形和结。我建议研究来自量子不变量的新的简单不变量的性质。特别地，我将研究将有色的琼斯多项式分解成一个无穷多个更简单的多项式序列，这些多项式似乎是亚历山大多项式的后代。

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.