Knotting and Linking Phenomena in Topology

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9803694 Cochran Dr. Cochran will investigate two areas. He will provide major new information about which knotted circles in 3-space are slice knots, i.e., bound embedded 2-disks in the 4-ball, and he will study invariants of 3-dimensional manifolds that can be computed algorithmically (invariants of finite type). As regards the first, in collaboration with Peter Teichner and Kent Orr, he has succeeded in finding an infinite sequence of geometric relations on knots that are successive approximations to the knots bounding an embedded disk in the 4-ball. These relations can be expressed in terms of both Freedman-Casson-Whitney towers and the language of surgery theory. He will investigate algebraic invariants that reflect these relations. He will provide a bridge between surgery theory, the techniques of Freedman-Casson, and the invariants of Casson and Gordon. He will also produce the first examples of knots that have vanishing Casson-Gordon invariants but are not slice knots. In the second area, in collaboration with Paul Melvin, he will focus on applications of their new theory of finite type invariants to existing problems about 3-manifolds, and to finding the topological significance of specific new invariants that he has defined from the quantum invariants. Dr. Cochran will study complex new phenomena of the knotting and linking of spatial objects. The study of the complex 3-dimensional geometric "shapes" assumed by (for example) closed DNA, carbon compounds, viruses and proteins is central to several frontiers of mathematical biology and pharmacology. For example, if the 3-dimensional structure of a virus is known and if a protein can be found with a complementary shape and chemistry, then this protein can be used as the basis of an anti-viral drug. Much of such research proceeds in an ad hoc fashion because of our ignorance of the 3-dimensional geometry of proteins. Dr. Cochran will focus on very new, "higher order" knotting phenomena. He seeks new geometric and quantitative descriptions of phenomena. ***