On Milnor's invariants of four-component links.
This paper studies three and four component links. The components are submanifolds of the link and need not be connected or non-empty. Two and three component links are semibounding if for any pair of connected components the linking number is \(0\), for a four component link there is an added condition that for any triple of connected components the integer-valued Milnor \(\mu\)-invariant is \(0\). Bordism in this class is defined and requires preserving the semibounding property under each surgery. The \(\mu\)-invariants are defined in terms of Seifert surfaces of the components of the link. The effect of a \(\Delta\)-move, defined by Murakami and Nakanishi, on a \(\mu\)-invariant is determined. For four-component links the \(\mu\)-invariants are defined, again in terms of Seifert surfaces. A balanced \(\Delta\)-move is defined as a pair of \(\Delta\)-moves, each involving three arcs, with an added condition which preserves the semibounding property when applied to a four-component semibounding link. The outline of a proof is given showing that one can get from any semibounding four-component link to another by a sequence of Reidemeister moves, \(\Delta\)-moves involving at most two components of the link, and balanced \(\Delta\)-moves. By determining the effect of these moves on the \(\mu\)-invariants, an axiomatic definition of the invariants is given.
scientific article; zbMATH DE number 1928189