On Gusarov's groups of knots

We give a construction of Gusaros 's groups G(n) of knots based on pure bra id commutators, and show that any element of G(n) is represented by an infi nite number of prime alternating knots of braid index less than or equal to n + 1. We also study V-n, the torsion-free part of G(n), which is the grou p of equivalence classes of knots which cannot be distinguished by any rati onal Vassiliev invariant of order less than or equal to n.. Generalizing th e 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 g roup.