Mathematical Sciences: Polynomial Invariants in Knot Theory

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This research centers on polynomial invariants in knot theory, their structure and implications for topology, and their interrelationships with mathematical physics. In particular, the project concentrates on the investigation of state models for knot polynomials. State models were first introduced by the principal investigator for the Alexander-Conway polynomial, and for the Jones polynomial. The project investigates these models and related models (Yang-Baxter models) involving statistical mechanics and quantum groups, and their relations with more general state models for the skein polynomial (Homfly and Kauffman) that are combinatorial in nature. The principal investigator intends to use state models to explore topological problems and to explore their connection with possible intrinsic (diagram-free) definitions of these invariants. Some fairly simple algebra concerning polynomials in one and two variables has found application to the problem of distinguishing between essentially different knots (as opposed to merely different presentations of the same knot). Simple though the idea is, important applications exist to the coiling and possible knotting of the genetic material DNA, as well as to problems in statistical mechanics.