The Geometry and Topology of Quantum Invariants of Knots and 3-Manifolds

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This project concerns the quantum invariants of 3-manifolds and links in them. A main problem is that these invariants are poorly understood from the point of view of topology and geometry. The goal of this project is to make connections between the quantum invariants and the existing knowledge of the geometry and topology of 3-manifolds. The main approach will be an investigation of the large level asymptotic expansion of the quantum invariants of closed 3-manifolds. The idea of studying these asymptotics stems from Witten's work on the quantum invariants. From that work and subsequent work of mathematicians there have emerged some rigorous mathematical conjectures about the geometric content of these asymptotics. These conjectures indicate that the quantum invariants hide deep topological information. The PI plans to check, in collaboration with others, (some of) the conjectures via concrete case studies. E.g., he will jointly with J.E. Andersen continue their project on the surgeries on the figure 8 knot. All but a finite number of these 3-manifolds are hyperbolic and it is the hope that this case study can shed some light on the connection between quantum invariants and hyperbolic geometry. A main question is whether the quantum invariants can detect the volume of hyperbolic 3-manifolds. In addition the PI will try to work from a more general point of view, namely he will apply T. Yoshida's new approach to the quantum invariants to try to calculate the asymptotic expansion of the invariants for all closed 3-manifolds, Yoshida's construction being based on an abelianization of conformal field theory. In that connection the PI will jointly with Yoshida work on proving that Yoshida's invariants coincide with the more well established Reshetikhin--Turaev invariants thereby making a connection to the other approaches to the quantum invariants.This project tries to develop new techniques to obtain knowledge about the shape (topology) and geometry of low-dimensional objects such as knots and 3-dimensional spaces. A main reason that we should study low-dimensional spaces and their geometry and shape is that nature contains such spaces in many contexts. Thus large molecules, e.g. DNA, have a "knotted" structure and newer research points in the direction that one can apply knot invariants to obtain information about such molecules. Certain of their properties depend on their shape, e.g., how they are "knotted". Another example is in cosmology where a central question is: What is the "shape" of our universe? Here one should think of the fact that when we walk around on the earth it just looks like an ordinary plane but in fact the surface of the earth is like the surface of a very large ball. It is similar with our universe. Locally everything looks flat, i.e., locally our universe looks like a standard 3-dimensional Euclidean space (a 3-dimensional pendant to a plane), but maybe the universe as a whole is something completely different, like a curved compact space. Mathematicians use so-called invariants to detect topological and geometric properties of spaces, e.g., if they are curved like a sphere or not like a plane. The quantum invariants, the theme for this project, is a relatively new family of invariants dating back to V. Jones' discovery of the Jones polynomial in the 80ties.