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.

The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S 3 − K , considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the knot group, G, these invariants reflect the structure ofterm of the derived series of G). Hence any phenomenon associated to Gis invisible to abelian invariants. This paper begins the systematic study of invariants associated to solvable covering spaces of knot exteriors, in particular the study of what we call the n th higher-order Alexander module,We show that these modules share almost all of the properties of the classical Alexander module. They are torsion modules with higher-order Alexander polynomials whose degrees give lower bounds for the knot genus. The modules have presentation matrices derived either from a group presentation or from a Seifert surface. They admit higher-order linking forms exhibiting self-duality. There are applications to estimating knot genus and to detecting fibered, prime and alternating knots. There are also surprising applications to detecting symplectic structures on 4-manifolds. These modules are similar to but different from those considered by the author, Kent Orr and Peter Teichner and are special cases of the modules considered subsequently by Shelly Harvey for arbitrary 3-manifolds.

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.

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In 1997, T. Cochran, K. Orr, and P. Teichner [12] defined a filtration of the classical knot concordance group C,