On Milnor's Invariants of 4-Component Links

We study the behavior of Milnor's $\mu$-invariants of three- and four-component links with respect to the discriminant determined by $\Delta$-moves of links. We introduce a new type of $\Delta$-move, balanced $\Delta$-moves, or, briefly, $B\Delta$-moves. Since each four-component link is equivalent to a standard link under a sequence of balanced $\Delta$-moves, $\Delta$-moves that involve at most two components, and Reidemeister moves, we manage to define axiomatically $\mu$-invariants of length 3 for arbitrary semibounding links.