Knot concordance, Whitney towers and L^2 signatures

TL;DR

This paper introduces a new geometric filtration of the knot concordance group using Whitney towers, constructs novel obstructions to sliceness, and employs L^2 signatures to detect non-slice knots beyond classical invariants.

Contribution

It develops a Whitney tower-based filtration, creates new obstructions in L-theory, and applies L^2 signatures to identify non-slice knots undetectable by previous invariants.

Findings

Constructed non-slice knots indistinguishable from slice knots by classical invariants.

Defined a geometric filtration of the knot concordance group using Whitney towers.

Introduced new obstructions in L-theory that vanish on slice knots.

Abstract

We construct many examples of non-slice 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 Casson-Gordon 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 {\em 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.