Clasper Concordance, Whitney towers and repeating Milnor invariants
Abstract
We show that for each k∈N, a link L⊂ S3 bounds a degree k Whitney tower in the 4-ball if and only if it is Ck-concordant to the unlink. This means that L is obtained from the unlink by a finite sequence of concordances and degree k clasper surgeries. In our construction the trees associated to the Whitney towers coincide with the trees associated to the claspers. As a corollary to our previous obstruction theory for Whitney towers in the 4-ball, it follows that the Ck-concordance filtration of links is classified in terms of Milnor invariants, higher-order Sato-Levine and Arf invariants. Using a new notion of k-repeating twisted Whitney towers, we also classify a natural generalization of the notion of link homotopy, called twisted self Ck-concordance, in terms of k-repeating Milnor invariants and k-repeating Arf invariants.