Whitney tower concordance of classical links

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. 57M25, 57M27, 57Q60; 57N10

Cochran's βi-invariants via twisted Whitney towers

“…have a non-zero relative Euler number ω(W J ) ∈ Z with respect to the standard framing of its boundary (e.g. [5,Sec.2.2]). This integer ω(W J ) is called the twisting of W J , and if ω(W J ) = 0, then a Whitney move guided by W J will create new self-intersections, even if W J happens to be embedded.…”

“…As per Definition 1.6 below, twisted Whitney disks are allowed in twisted Whitney towers (e.g. [5,6]), while earlier papers defined Whitney towers with the requirement that Whitney disks be framed. For conversational ease we may sometimes let "Whitney tower" refer to either twisted or framed Whitney towers in a general discussion that applies to both settings, hopefully when no confusion will result.…”

“…More precisely, we showed in [6] how the first non-vanishing µ-invariants can be computed from t(W) modulo certain relations, all of which can be realized via geometric maneuvers preserving the order of W (without changing its boundary L). Most prominently, the geometric IHX-relations, or 4-dimensional Jacobi-identities, can be used to change t(W) by replacing a tree containing an I-shaped subtree with two trees of the same order that only differ locally by H-and X-shaped subtrees, plus a number of trees of higher order [5]. For instance, at the cost of creating higher-order trees, the geometric IHX-relations can be used to modify an order n twisted Whitney tower so that all framed trees in t(W) with two 2-labels and n 1-labels are isomorphic to t n in Figure 1, and all twisted trees with one 2-label and n 2 1-labels are isomorphic to t n 2 if n is even.…”

Concordance of links with identical Alexander invariants

“…In this section, we observe that our links L i in Theorem B can be assumed to be mutually order n Whitney tower/grope concordant for any n. For the definition of order n Whitney tower concordance of framed links, see [13,Definition 3.2]; in this section, we assume that links are always endowed with the zero-framing.…”

“…We remark that the links L i in Theorem B can be chosen in such a way that they are indistinguishable to the eyes of the asymmetric Whitney tower/grope theory, which is another framework for the study of link concordance extensively investigated in recent work of Conant, Schneiderman, and Teichner (see [12] as an extended summary providing other references). Namely, the L i (with the zero-framing) are mutually order n Whitney tower/grope concordant for any n in the sense of [13,Definition 3.1]. This is discussed in Section 5.…”