Invariants of Knots and 3-Manifolds

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9800703 Garoufalidis The investigator's recent and current research is on the notion of finite type invariants of links and 3-manifolds. The motivation for the project comes from several sources: (i) perturbative Chern-Simons theory, (ii) generalizing the notion of Vassiliev invariants of knots in the 3-sphere, (iii) separating the quantum group invariants of knots and 3-manifolds into elementary pieces and reassembling them. One of the goals of the theory of finite type invariants of knots and 3-manifolds involves understanding the role and the strength of these invariants in the classification of knots and 3-manifolds. Thirteen years ago, Vaughan Jones discovered a simple yet remarkably useful polynomial associated to every knot in 3-dimensional space. Shortly afterwards, Edward Witten studied a 3-dimensional gauge field theory that generalized the Jones polynomial to knots in arbitrary 3-dimensional space. Gauge theories in 3 and 4 dimensions are intimately related to the high energy description of our physical world (in the energies and the distances that we currently perceive). In addition, knots appear naturally as supercoiled forms of DNA (an important ingredient of life), as well as being trajectories of some elementary particles at high energies. Thus, the current research project has potential applications, yet to be fully understood, in biology and in high energy physics. ***