Whitehead double and Milnor invariants

TL;DR

This paper investigates how Whitehead double operations affect Milnor invariants of links, revealing that such operations can double the length of vanishing invariants and establishing connections between link-homotopy and self Delta-equivalence.

Contribution

It provides new formulas for Milnor invariants after Whitehead doubling and links link-homotopy with self Delta-equivalence through this operation.

Findings

Whitehead double doubles the vanishing length of Milnor invariants.

Formulas for the first non-vanishing Milnor invariants after Whitehead doubling.

Characterization of link-homotopy to the unlink via Whitehead double and self Delta-equivalence.

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 < k are all zero into a link with vanishing Milnor invariants of length < 2k, and we provide formulas 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 a link L with a single component Whitehed doubled is self Delta-equivalent to the unlink.