Mathematical Sciences: Properties of Quantum Invariants in 3-Dimensional Topology

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9704893 Rozansky An important development in 3d topology of the last decade was a discovery of "quantum invariants," most notably, the Jones polynomial of links and the Reshetikhin-Turaev (RT) invariant of 3-manifolds. Although these invariants are quite effective in classifying knots and 3-manifolds, their topological nature remains mostly obscure. The purpose of this research is to study the topological origin of quantum invariants by decomposing them into simpler pieces, called "finite type invariants," and trying to explain the nature of these pieces individually. This might be accomplished by using the tools of quantum field theory, in particular, the asymptotic expansion of path integrals through Feynman diagrams. E. Witten has identified the Jones polynomial and RT invariant as certain path integrals taken over the classes of SU(2) connections on a 3-manifold. This approach has already led to a discovery of the Alexander polynomial and Milnor linking numbers inside the Jones polynomial. The simplest "pieces" of the RT invariant were identified (at least, as a conjecture) with the Chern-Simons invariant of flat connections, Reidemeister torsion and SU(2) Casson invariant. One hopes that more complicated topological invariants, such as the Casson invariant of other Lie groups, will be found among other pieces of the RT invariant. A classification of knots is an open topological problem. A knot in topology is a closed rope (i.e., a circle) which is knotted. Can a particular knot be untangled by continuously deforming the rope without cutting it? How can one determine this just by looking at a picture of the knot? This problem may be solved if one finds enough knot invariants. A knot invariant is a number that can be calculated by examining a picture of a knot. The number should not change when a knot is continuously deformed. Thus if a knot invariant takes different values on two knots, then these knots are different, because they cannot be deformed into one another. For a long time the only effective knot invariant was the Alexander polynomial. A lot of new invariants, such as the Jones polynomial, were discovered during the last decade. These invariants are quite effective in distinguishing knots, but their origin is still a mystery, because their topological interpretation is mostly missing. The purpose of this project is to try to fill this gap by using the tools of quantum field theory, such as path integral and Feynman diagrams. The relevance of quantum field theory to low dimensional topology was discovered by E. Witten. He demonstrated that the Jones polynomial comes from a particle theory in which knots appear as space-time trajectories of hypothetical elementary particles. The particles interact with each other in ways that resemble the "real life" particles. This approach has yielded numerous mathematical conjectures that may lead to a better understanding of the topological nature of the Jones polynomial and ultimately to the classification of knots. ***