Mathematical Sciences: Polynomial Invariants in the Theory of Knots

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9504471 Kauffman This project investigates invariants of knots, 3-manifolds and higher dimensional manifolds that arise from state summation models, quantum groups and topological quantum field theory. This includes work on the structure of the Jones polynomial, on Vassiliev invariants, and on connections with quantum field theory and molecular biology, as well as techniques and conjectures about the use of functional integration in topology. In the last decade, knot theory has encountered an awakening through the infusion of a remarkable collection of techniques from mathematical physics and a series of interrelationships with molecular biology, chemistry, and theoretical physics. The principal investigator is one of the key figures in the inception of this relationship with physics (statistical mechanics) via his introduction of the use of state summation models for polynomial invariants of knots and links. In a state summation model the knot is regarded as a miniature physical system, and topological properties of the knot are seen as averages of physical interactions. These models have grown to encompass techniques from quantum field theory and have been useful in topology and its applications to the study of molecular interactions in biology and chemistry. In topology these techniques apply to the structure of three- and four-dimensional spaces as well as to knots. This means that these techniques are essential to the architecture of three- dimensional structures - from molecules to networks of communication, from atoms to galaxies. All fields of science are based on the structure of space. What is particular to this project's topological study is that it articulates the topological components in large networks of which knots are the principal example. Such networks are the basis of significant structures such as RNA and DNA, where topology is the key to interaction. In fact, the introduction of topology in these subjects is, from the viewpoint of this principal investigator, just the beginning of a larger interplay, where topology is seen as the basis for studying the recursive and circular architectures of complex systems. A galaxy, a society, an organism and a physical system all have in common the stability that arises through closed systems of circular interactions. Such interactions have traditionally been treated via cybernetics and computer modeling. These topological investigations add a new dimension to cybernetics and complex systems, providing new questions and a host of new problems. All these issues are treated in the new book "Knots and Applications," edited and contributed to by the principal investigator. ***