Lifting Milnor Invariants for 3-Component Links

Abstract

We define a sequence of integer-valued invariants $γ^k(L)$ for a $3$-component link $L$. We prove that the resulting ${\hat{I}}^3$-invariants are invariant under concordance, and more generally under weak cobordism, and that they lift certain Milnor invariants of 3-component links. To establish this, we introduce an invariant $h(L)$, a $3$-component analogue of the Kojima--Yamasaki $\hat{I}\cdot$-invariant, and show that it recovers the ${\hat{I}}^3$-invariants. As applications, we obtain a weak-cobordism classification when the distinguished component has trivial Alexander polynomial and characterize knots that bound continuously embedded disks in $B^4$ whose complements have fundamental group $\mathbb{Z}$.

30 pages, 7 figures. Version 2: Revised to explain some connections to work of Tatsuya-Yasuhara