# Shake Slice and Shake Concordant Links

## TL;DR

This paper introduces the concept of shake slice and shake concordance for links, extending previous knot-based notions to links with multiple components, and provides classifications and invariants related to these concepts.

## Contribution

It generalizes shake slice and shake concordance to links, introduces new notions like strongly shake slice, and characterizes these in terms of classical invariants and concordance.

## Key findings

- Infinite families of links distinguish shake concordance from concordance.
- Complete characterization of shake slice and shake concordance for r=0.
- Milnor invariants are invariants of shake concordance.

## Abstract

We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake r-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly r-shake slice and strongly r shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for r=0 we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor mu bar invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.