Milnor’s invariants and self $C_{k}$-equivalence

Abstract

It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor’s invariants with repeated indices are invariants not only of isotopy, but also of self $C_{k}$-equivalence. Here self $C_{k}$-equivalence is a natural generalization of link homotopy based on certain degree $k$ clasper surgeries, which provides a filtration of link homotopy classes.