Whitehead double and Milnor invariants

Abstract

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 $\leq k$ are all zero into a link with vanishing Milnor invariants of length $\leq 2k+1$, and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self $\Delta$-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 doubled is self $\Delta$-equivalent to the unlink.