Knots, von Neumann signatures, and grope cobordism

Abstract

We explain new developments in classical knot theory in 3 and 4~dimensions, i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In dimension~3 we give a geometric interpretation of the Kontsevich integral (joint with Jim Conant), and in dimension 4 we introduce new concordance invariants using von Neumann signatures (joint with Tim Cochran and Kent Orr). The common geometric feature of our results is the notion of a grope cobordism.