The geometry of the knot concordance space

Abstract

Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi-n-flats for every n, implying that C admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasi-homomorphism P: C --- C with respect to the metric given by the slice genus. We show that winding number one satellite operators induce quasi-isometries. We prove that strong winding number one satellite operators induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasi-contractions. These results contribute to the conjecture that C is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators.