2+1 dimensional Topological Quantum Field Theories a la Yoshida

This project concerns the study of knots, links, and other low-dimensional objects. It is a mathematical project, but one of its central components is to develop techniques and mathematical quantities which are applicable to problems in modern biochemistry such as the investigation of mechanisms behind viruses (more of this below).

The project involves the study of a certain family of link invariants called quantum invariants. The focus will be on obtaining a deeper knowledge about these invariants' ability to separate knots according to their topological type. To obtain such knowledge, research points in the direction that it is crucial to have a geometric understanding of quantum invariants. Today we are far from having such understanding. It is of crucial importance to know a given invariants ability to distinguish knot types if one wants to apply this invariant in an efficient way to biochemistry.

There are different approaches to the quantum invariants. In this project we will focus on a new approach due to Tomoyoshi Yoshida, the Department of Mathematics at the Tokyo Institute of Technology, Japan. We believe for various technical reasons that his approach is particularly well suited to obtain a deeper geometric understanding of the quantum invariants.
The complement of a given knot is the surrounding 3-dimensional space of the knot, i.e. the space left after removing a 'small' tubelike neighborhood of the knot.
It turns out that we can study knots by studying their complements. Thus the study of knots can
be seen as a part of the study of 3-dimensional spaces.
So far Yoshida has constructed invariants of so-called closed 3-manifolds. Extensions of his theory are necessary to allow for boundaries which is necessary if we want to use his approach to study knots and 3-manifolds efficiently.
Thus our first goal is to extend Yoshida's theory to a so-called 2+1 dimensional topological quantum field theory.
Another goal is to calculate and geometrically interpret a certain asymptotic expansion of the invariants (the so-called large quantum level asymptotics). Certain calculations point in the direction that these asymptotics contain a wealth of geometric information about the quantum invariants. These results have led to a list of conjectures about these asymptotics. In the best case a full understanding of the large level asymptotics could give us a complete description of the quantum invariants ability to separate different knot types.

Let us end by returning to viruses. Viruses attack cells in order to alter the DNA inside them. To do this, they bring closer certain parts of the DNA, then cut them and stick them back together differently in such a way that the molecule of DNA is transformed into a knot. One of the essential aspects of the struggle against viruses is to recognize the signature of different viruses by their effects on the DNA. One can characterize these effects by the topological (isotopy) type of knot which results from the action of the virus. But then it is necessary to be able to recognize the knot in question if one wants to find out which virus it is. The job is then to implement applications of suitable quantum invariants as knot detectors, but then it is crucial to know how good these invariants are as knot detectors.

The project has received partial funding from the
National Science Foundation (NSF) in USA, DMS-0604994.