Knot Theory and Geometry of Manifolds

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It is well-known that in order to study a geometry of manifold, it is important to study the topological structure. The study of topology is to analyze the position and the shape of a topological object, and the study of position is to analyze a pair of manifold and submanifold, represented typically by knot theory. Knot theory and related topics are studied actively for the last two decades not only abroad but also much more in Japan. During this research program, knot theory and the related studies of low dimensional manifolds have been studied by many researchers. For example, Kawauchi obtained a new concept "exact 4-manifold" by studying a surface-knot. This concept is useful to classify 4-manifolds with infinite cyclic first homology, and as a result, we see that there exists a surface-knot invariant which is analogous to the Arf invariant of a classical knot. In other related studies, hyperbolic geometry, differential topology (including handle theory, Morse theory), gauge theory, transformation theory, foliation theory, homotopy theory, real and complex singularities, dynamical systems, general topology, surface moduli have been studied. Also, a "mew applied knot theory" was searched in relations with Yang-Baxter equation (in statistical mechanics), a molecular graph (in molecular chemistry), and DNA knot (in biochemistry).