Whitehead double and milnor invariants

We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length ≤ k are all zero into a link with vanishing Milnor invariants of length ≤ 2k +1, and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Δ-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if the link L with a single component Whitehead 1 doubled is self Δ-equivalent to the unlink.