Untying Knots in 4-Dimensional Space

Tim Cochran (Rice University)

ABSTRACT Topology is the area of mathematics concerned with the "shape" of geometrical objects. Shape is very important in the study of networks, in the design of drugs, in satellite recognition, and in the mapping of human organs. Within Topology, Knot Theory is the study of knotted circles in 3-dimensional space. A mathematical knot can be obtained from a piece of rope by tying an ordinary knot into it and then fusing together the two free ends. In this talk I will introduce knot theory via an open problem first posed in the 1950's because of its relation to classifying 4-dimensional manifolds: "Which knots can be untied in 4-dimensional space?"