Milnor invariants of covering links

TL;DR

This paper introduces a generalization of Milnor invariants for covering links, demonstrating their invariance under cobordism and their ability to distinguish links where traditional invariants fail, with applications to Brunnian links.

Contribution

It defines Milnor invariants for covering links, proves their cobordism invariance, and relates them to classical invariants for Brunnian links, extending the understanding of link invariants.

Findings

Milnor invariants for covering links are cobordism invariants.

These invariants can distinguish links with identical ordinary Milnor invariants.

Sum of linking numbers of covering links relates to Milnor invariants modulo 2.

Abstract

We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invariants coincide. Moreover, for a Brunnian link $L$, the first non-vanishing Milnor invariants of $L$ is modulo-$2$ congruent to a sum of Milnor invariants of covering links. As a consequence, a sum of linking numbers of ' iterated' covering links gives the first non-vanishing Milnor invariant of $L$ modulo $2$.