Knot concordance and higher-order Blanchfield duality
The authors introduce new techniques for studying the knot concordance group \(\mathcal C\) and apply these to resolve a long-standing conjecture about whether certain natural families of knots contain nonslice knots. The conjecture referred to above involves recursive families \(J_n\) of algebraically slice knots modeled on an initial knot \(J_0\) using a banding construction. For a knot \(K\) and a point in the circle, there is an associated Levine-Tristram signature, which then gives a signature function defined on the circle. Let \(\rho_0(K)\) denote the integral of this function over the unit circle, normalized to have length 1 -- this is an average Levine-Tristram signature for \(K\). The authors show: There is a constant \(C\) such that if \(|\rho_0(J_0)| > C,\) then for each \(n \geq 0, \, J_n\) is of infinite order in the topological and smooth knot concordance groups. Furthermore, there is a constant \(D\) such that if \(J_2\) is a slice knot then \(\rho_0(J_0) \in \{0,D\}.\) This result was classically known only for \(n = 0,1,\) using Levine-Tristram signatures and Casson-Gordon invariants respectively. The constant \(D\) is a special real number associated to the \(9_{46}\) knot and may be 0. Another major result of the paper is to show that each quotient \(\mathcal F_n/\mathcal F_{n.5}\) in the Cochran-Orr-Teichner filtration \(\{\mathcal F_n\}\) of \(\mathcal C\) has infinite rank. This was previously known only for \(n = 0,1,2.\) The construction of the examples is done in the smooth category and so establishes these results on the quotients in the smooth category as well. One facet of the techniques involved is that they are simpler than earlier techniques used by \textit{T. D. Cochran} and \textit{P. Teichner} [Duke Math. J. 137, No. 2, 337--379 (2007; Zbl 1186.57006)] to show that the quotients had rank at least one. The knots \(J_n\) (for suitable \(J_0\)) are used in the proof.
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