Topologically slice knots of smooth concordance order two. (English) Zbl 1339.57011

Adding to knowledge about topologically slice, but not smoothly slice, knots, the authors prove that the subgroup of topologically slice knots in the Fox-Milnor smooth concordance group contains a subgroup isomorphic to \(\mathbb Z_{2}^\infty\) with the property that no non-trivial element of that subgroup has a representative with Alexander Polynomial equal to \(1\).
The proof is explicit, constructive, occupies 30 pages and references many papers. It makes use of some number theory, 9 papers of Ozsváth and Szabó and Heegard-Floer Theory. Of particular importance is an invariant (\(d\)-invariant) measuring obstruction to slicing. The invariant was first introduced in [C. Manolescu and B. Owens, Int. Math. Res. Not. 2007, No. 20, Article ID rnm077, 21 p. (2007; Zbl 1132.57013)], and used in [J. E. Grigsby et al., Geom. Topol. 12, No. 4, 2249–2275 (2008; Zbl 1149.57007)] and [S. Jabuka and S. Naik, ibid. 11, 979–994 (2007; Zbl 1132.57008)].


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R58 Floer homology
53D40 Symplectic aspects of Floer homology and cohomology
57N70 Cobordism and concordance in topological manifolds
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