Knot concordance, Whitney towers and \(L^2\)-signatures. (English) Zbl 1044.57001

A knot \(K\) in the 3-sphere is slice if there exists a locally flat topological embedding of the 2-disk into \(B^4\) whose restriction to the boundary is \(K\). Two knots are concordant if there is a locally flat topological embedding of the annulus into \(S^3 \times [0,1]\). The set of concordance classes form an abelian group under the operation of connected sum of knots \(\mathcal C\), whose identity element is the class of slice knots. In the article under review a geometrically defined filtration of the concordance group \(\mathcal C\) is exhibited, indexed by the half integers \[ \dots \subset {\mathcal F}_{n.5} \subset {\mathcal F}_n \subset \dots \subset {\mathcal F}_{0.5} \subset {\mathcal F}_0 \subset {\mathcal C}. \]
The following facts are established about the groups \({\mathcal F}_h\), \(h\in 1/2 {\mathbb{N}}_0\): (i) A knot is in \({\mathcal F}_0\) if and only if it has a vanishing Arf invariant. (ii) A knot is in \({\mathcal F}_{0.5}\) if and only if it is algebraically slice. (iii) If a knot is in \({\mathcal F}_{1.5}\) then all previously known concordance invariants (such as the Casson-Gordon invariants) vanish. (iv) The quotient group \({\mathcal F}_2/{\mathcal F}_{2.5}\) has infinite rank.
Moreover the authors conjecture that for any \(n\), \({\mathcal F}_n/{\mathcal F}_{n.5}\) has infinite rank. Finally, the authors exhibit a class of infinitely many knots in \({\mathcal F}_2\) that have vanishing Casson-Gordon invariants but are not topologically slice.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N70 Cobordism and concordance in topological manifolds
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