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

Hedden, Matthew; Kim, Se-Goo; Livingston, Charles

J. Differ. Geom. 102, No. 3, 353-393 (2016).

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)].

Reviewer: Lee P. Neuwirth (Princeton)

MSC:

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

Keywords:

knot concordance group; Fox-Milnor group; topologically slice knot; smooth slice knot; Heegard Floer theory; spin structure

Citations:

Zbl 1132.57013; Zbl 1149.57007; Zbl 1132.57008