In [Adv. Math. 173, 179–261 (2003; Zbl 1025.57016)], P. Ozsváth and Z. Szabó constructed an invariant \(d\) of rational homology \(3\)-spheres endowed with a Spin\(^{\text{c}}\) structure. Given a knot \(K\) in the \(3\)-sphere, one can construct its \(2\)-fold branched cover, which is a rational homology \(3\)-sphere admitting a unique Spin\(^{\text{c}}\) structure. Ozsváth and Szabó’s invariant can thus be seen as an invariant \(\delta\) of the knot \(K\). It follows from the properties of \(d\) studied in the aforementioned paper, that \(\delta\) is a surjective group homomorphism from the smooth concordance group of knots in the \(3\)-sphere onto the integers, i.e. \(\delta\) is an invariant of knot concordance.
The authors compare \(\delta\) to other invariants of knot concordance, and prove that, for alternating knots, \(\delta\) coincides with half the opposite of the knot signature. The authors conjecture that this is true for a larger class of knots, namely \(H\)-thin knots, that is knots whose Khovanov homology is supported on two adjacent diagonals. The authors show by direct computation that \(\delta\) not always equal to half the opposite of the knot signature. Moreover, they exhibit an infinite family of knots for which \(\delta\) is a non-trivial obstruction to sliceness, while the other concordance invariants considered are not.
A sufficient condition for positive untwisted Whitehead doubles not to be smoothly slice is also established.