This project studies problems related to polynomial invariants in knot theory, particularly for the Jones polynomial and its generalizations via methods in statistical mechanics, quantum groups and quantum field theory. A central problem about the Jones polynomial is whether its triviality implies the triviality of the knot to which it is applied. An affirmative answer to this question would constitute a breakthrough in our understanding of the nature of knottedness in three-dimensional space. Mathematical ideas motivated by physical ideas have succeeded in producing a medley of invariants of knots, links, and three-manifolds. The investigator is in the process of extending these ideas to knotted surfaces in four-dimensional space and to four-dimensional manifolds. Here the discovery of new invariants would itself constitute a breakthrough in the study of four dimensions. Such a breakthrough would have implications for the physics of quantum gravity. The mathematical study of knots is important not only for its own sake, but also for its applications to molecular biology and its relationships with physics and the foundations of mathematics. The research indicated in this project is linked with these applications and with the use of both computer graphics and expository writing to enlarge the discussion of these ideas in both interdisciplinary and educational contexts. By using computer models of knots, one can study questions such as how the knot will behave if it is coated with electrical charge (self-repelling knots), how it will appear if it is tied with a least amount of rope for a given diameter, how the geometry of the complement of the knot appears to an observer flying about in it, how DNA molecules behave when moving in their cellular environment. All these directly physical questions are, at levels that we do not yet fully understand, related to the more algebraic methods of the polynomial invariants. Knots and Nature are inextricably entwined. ***
