  To a special type of grope embedded in 4-space, that we call an admissible
grope, we associate a length function for each real number q at least 1. This
gives rise to a family of pseudo-metrics d^q, refining the slice genus metric,
on the set of concordance classes of knots, as the infimum of the length
function taken over all possible grope concordances between two knots. We
investigate the properties of these metrics. The main theorem is that the
topology induced by this metric on the knot concordance set is not discrete for
all q>1. The analogous statement for links also holds for q=1. In addition we
translate much previous work on knot concordance into distance statements. In
particular, we show that winding number zero satellite operators are
contractions in many cases, and we give lower bounds on our metrics arising
from knot signatures and higher order order signatures. This gives further
evidence in favor of the conjecture that the knot concordance group has a
fractal structure.
