  We show that two knots have matching Vassiliev invariants of order less than
n if and only if they are equivalent modulo the nth group of the lower central
series of some pure braid group, thus characterizing Vassiliev's knot
invariants in terms of the structure of the braid groups. We also prove some
results about knots modulo the nth derived subgroups of the pure braid groups,
and about knots modulo braid subgroups in general.
