Skein relations for Milnor's $\mu$-invariants

Abstract

The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor’s $\bar{\mu }$–invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Alexander and Jones invariants are known for quite a while, a similar treatment of Milnor’s $\bar{\mu }$–invariants was missing. We fill this gap by deducing simple skein relations for link-homotopy $\mu$–invariants of string links.