Campbell-Hausdorff invariants of links

Abstract

We construct a sequence of concordance invariants for classical links, which depend on the peripheral isomorphism type of the nilpotent quotients of the link fundamental group. The terminology stems from the fact that we replace the Magnus expansion in the definition of Milnor's μ-invariants by the similar Campbell-Hausdorff expansion. The main point is that we introduce a new universal indeterminacy, which depends only on the number of components of the link. The Campbell-Hausdorff invariants are new, effectively computable and can efficiently distinguish (unordered and unoriented) isotypes of links, as we indicate on several families of closed braid examples. They also satisfy certain natural dependence relations, which generalize well-known symmetries of the μ-invariants.