Whitney tower concordance of classical links

TL;DR

This paper advances the understanding of classical link concordance by computing Whitney tower filtrations, introducing new invariants, and providing geometric characterizations of Milnor invariants, thereby deepening the intersection theory of links.

Contribution

It introduces a quadratic refinement for twisted Whitney towers and classifies Whitney tower filtrations using Milnor, Sato-Levine, and Arf invariants, expanding the theoretical framework.

Findings

Whitney tower filtrations are finitely generated abelian groups.

New higher-order Sato-Levine and Arf invariants are identified.

Complete classification of filtrations via classical and new invariants.

Abstract

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. Applications include computation of the grope filtration, and new geometric characterizations of Milnor's link invariants.