An 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.
