We develop our general
machinery of the Campbell–Hausdorff invariants of links, in the case of pure links,
with emphasis on the connections with the lower central series of the pure braid
groups. We present a complete simple set of rules for the Artin calculus of longitudes
modulo the central series. We prove that if two pure links differ by an order k pure
braid commutator, then their order k Campbell–Hausdorff invariants p(k) are the
same. In this case, the general theory offers a decision test for the equality of
p(k+1)-invariants. We introduce the notion of homogenous link, which leads
to important computational improvements for the general p(k+1)-test. We
provide both general homogeneity criteria and concrete interesting classes
of homogenous examples. We illustrate the efficiency of our approach, on
several classes of examples which cannot be distinguished by other known link
invariants.
