A Quantum Field Theory Approach to the Study of Low-dimensional Topology Invaraints and their Categorification

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

The Jones polynomial invariant of links presents two puzzles. The first one is its interpretation within the framework of classical topology and its relation to the fundamental group of the link complement. This relation is well-known for the Alexander polynomial, but its Jones analog is missing. I search for it by using the quantum field theory approach. I decompose the colored Jones polynomial in the semi-classical limit into a sum of many simpler invariants, which turn out to be multi-variable polynomial invariants of links. These invariants are arranged naturally into a `tower' and the Alexander polynomial lies at its foundation. My goal is to study the properties of the other `higher level' polynomials and establish their relation to the topology of the knot complement. The second puzzle of the Jones polynomial is its polynomial structure, which does not follow easily from its Chern-Simons-Witten path integral presentation. An amazing explanation for the polynomial nature of the Jones polynomial comes from Khovanov's categorification program, which interprets the Jones polynomial as a graded Euler characteristic of a chain complex of graded modules, associated to a link up to a homotopy. I will study the possible quantum field theory interpretations of the Khovanov homology. Namely, the homology of the categorification complex should be a space of states for a 4-dimensional topological quantum field theory, which has to be constructed. This construction should help us to extend Khovanov's results to quantum polynomial invariants of 3-manifolds and to interpret their relation to 4-dimensional topology. The problem of knot classification is very old and although its formulation is simple and transparent, it has not been solved yet despite concerted efforts of many mathematicians. One of the most successful approaches to knot classification is the construction of knot invariants, that is, the numbers, which can be easily computed by looking at a picture of a knot and which would be the same for all pictures representing the same knot. This area has undergone significant developments in the last 30 years starting with the invention of the Jones polynomial. This polynomial and the other invariants which followed it, are intimately related to quantum field theory and can be expressed as so-called path integrals. Thus many of the new ideas in the theory of knot invariants are inspired by methods and approaches of quantum physics. The goal of my research is to use the methods of quantum field theory in order to interpret the new `quantum' topological invariants within the framework of classical topology and to apply them to the solution of the knot classification problem.