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The reduced peripheral system was introduced by Milnor in the fifties for thestudy of links up to link-homotopy, i.e. up to isotopies and crossing changeswithin each link component. However, for four or more components, thisinvariant does not yield a complete link-homotopy invariant. This paperprovides two characterizations of links having the same reduced peripheralsystem: a diagrammatical one, in terms of link diagrams, seen as weldeddiagrams up to self-virtualization, and a topological one, in terms of ribbonsolid tori in 4--space up to ribbon link-homotopy.

Characterization of the reduced peripheral system of links