Braid Commutators and Homogenous Campbell-hausdorff Tests

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 are the same. In this case, the general theory offers a decision test for the equality of p-invariants. We introduce the notion of homogenous link, which leads to important computational improvements for the general p-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.