A rational valued metric on the knot concordance group coming from gropes

Most of the 50-year history of the study of the set of smooth knot concordance classes, $\mathcal{C}$, has focused on its structure as an abelian group. A few years ago, Tim Cochran and I took a different approach, namely we studied $\mathcal{C}$ as a metric space (with the slice genus metric or the homology metrics) admitting many natural geometric operators, especially satellite  operators.  The hope was to give evidence that the knot concordance is a fractal space.  However, both of these metrics are integer valued metrics and so induce the discrete topology.  Here (with Mark Powell) we define a family of metrics, called the $q$-grope metrics, which take values in the real numbers.  We will show that there are sequences of knots whose $q$-norms get arbitrarily small for $q>1$.  We will also show that for any winding number 0 satellite operator, $R\colon \mathcal{C}\to \mathcal{C}$, is a contraction for $q$ large enough. This is joint work with Tim Cochran and Mark Powell.