Polynomial Invariants in the Theory of Knots

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9802859Kauffman This project studies problems related to polynomial invariants in knottheory, particularly for the Jones polynomial and its generalizations viamethods in statistical mechanics, quantum groups and quantum field theory.A central problem about the Jones polynomial is whether its trivialityimplies the triviality of the knot to which it is applied. An affirmativeanswer to this question would constitute a breakthrough in ourunderstanding of the nature of knottedness in three-dimensional space.Mathematical ideas motivated by physical ideas have succeeded in producinga medley of invariants of knots, links, and three-manifolds. Theinvestigator is in the process of extending these ideas to knotted surfacesin four-dimensional space and to four-dimensional manifolds. Here thediscovery of new invariants would itself constitute a breakthrough in thestudy of four dimensions. Such a breakthrough would have implications forthe physics of quantum gravity. The mathematical study of knots is important not only for its ownsake, but also for its applications to molecular biology and itsrelationships with physics and the foundations of mathematics. Theresearch indicated in this project is linked with these applications andwith the use of both computer graphics and expository writing to enlargethe discussion of these ideas in both interdisciplinary and educationalcontexts. By using computer models of knots, one can study questions suchas 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 leastamount of rope for a given diameter, how the geometry of the complementof the knot appears to an observer flying about in it, how DNA moleculesbehave when moving in their cellular environment. All these directlyphysical 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.***