We give a construction of Gusarov's groups
[Gscr ]n of knots based on pure braid
commutators, and show that any element of [Gscr ]n
is represented by an infinite
number of prime alternating knots of braid index less than or equal to
n+1. We also
study [Vscr ]n, the torsion-free part of [Gscr ]n,
which is the group of equivalence classes of
knots which cannot be distinguished by any rational Vassiliev invariant
of order less
than or equal to n. Generalizing the
Gusarov–Ohyama definition of n-triviality, we
give a characterization of the elements of the nth group of the
lower central series of an arbitrary group.