We construct many examples of nonslice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the CassonGordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain universal groups. Finally, we use the dimension theory of von Neumann algebras to define an L 2 -signature and use this to detect the first unknown step in our obstruction theory.

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Abstract. We provide new information about the structure of the abelian group of topological concordance classes of knots in S 3 . One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon invariants but whose non-triviality is detected by von Neumann signatures. Mathematics Subject Classification (2000). 57M25.

Abstract. We prove the nontriviality at all levels of the filtration of the classical topological knot concordance group C. This filtration is significant because not only is it strongly connected to Whitney tower constructions of Casson and Freedman, but all previously-known concordance invariants are related to the first few terms in the filtration. In [COT] we proved nontriviality at the first new level n = 3 by using von Neumann ρ-invariants of the 3-manifolds obtained by surgery on the knots. For larger n we use the Cheeger-Gromov estimate for such ρ-invariants, as well as some rather involved algebraic arguments using our noncommutative Blanchfield forms. In addition, we consider a closely related filtration, {G n }, of C defined in terms of Gropes in the 4-ball. We show that this filtration is also non-trivial for all n > 2.

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