## Osaka Journal of Mathematics

### 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.

#### Article information

Source
Osaka J. Math., Volume 48, Number 2 (2011), 371-381.

Dates
First available in Project Euclid: 6 September 2011

https://projecteuclid.org/euclid.ojm/1315318345

Mathematical Reviews number (MathSciNet)
MR2831978

Zentralblatt MATH identifier
1237.57007

#### Citation

Meilhan, Jean-Baptiste; Yasuhara, Akira. Whitehead double and Milnor invariants. Osaka J. Math. 48 (2011), no. 2, 371--381. https://projecteuclid.org/euclid.ojm/1315318345

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