Knot concordance, Whitney towers and L²-signatures

Abstract

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.

Structure in the classical knot concordance group

“…4.3]. (2.3) (subgroup property) If φ factors through a subgroup Γ , then ρ(M, φ) = ρ(M, φ 0 ) where φ 0 : π 1 (M ) −→ Γ is the induced factorization of φ [COT,Proposition 5.13]. This is a consequence of the corresponding fact for the canonical trace on a group von Neumann algebra.…”

“…We remark that if n-solvable is replaced by rationally n-solvable (see [COT,section 4]) then the analogous result holds without a condition on the Arf invariant of J. This is of interest in studying knots which bound disks in rational homology balls.…”

“…The reader who is not familiar with [COT] can see that the latter follows from the very believable fact that H 2 (W ; Z) = 0 implies that H 2 of the Γ-cover of W is Γ-torsion, hence H 2 (W ; K Γ ) = 0 [COT,Prop. 4.3].…”

“…It is sufficient to exhibit an infinite set of (2.0)-solvable knots K i , i ∈ Z + , such that no non-trivial linear combination is (2.5)-solvable. Let K r be the "seedling" ribbon knot shown in Figure 3.a and η be the designated circle just as was used in [COT,section 6]. Let J i be the knots of Proposition 2.6 and let K i = K r (J i , η) be the family of knots resulting from the grafting construction of Example 3.3.…”

“…and σ 0 is the ordinary signature [COT,Lemma 5.4,Def. 5.3] (in fact we prove in the Appendix that ρ(M, φ) is the integral of the Levine-Tristram signature function of the knot K normalized to have value 0 at z = 1, although this more precise fact is not strictly needed).…”