A cabling formula for the \(\nu^+\) invariant (Q2814427)

The subject of investigation is the Heegaard Floer based knot concordance invariant \(\nu^+\) for the cable knot \(K_{p,q}\) of a classical knot \(K\). The invariant \(\nu^+\) was introduced by \textit{J. Hom} and \textit{Z. Wu} [J. Symplectic Geom. 14, No. 1, 305--323 (2016; Zbl 1348.57023)] and independently, named \(\nu^-\), by \textit{P. Ozsváth} et al. [``Concordance homomorphisms from knot Floer homology'', Preprint, \url{arXiv:1407.1795}]. It is a lower bound for the \(4\)-ball genus \(g_4\) and as such an improvement upon the \(\tau\)-invariant of \textit{P. Ozsváth} and \textit{Z. Szabó} [Geom. Topol. 7, 615--639 (2003; Zbl 1037.57027)]. The main result is an equation \(\nu^+\!\!(K_{p,q}) = p \nu^+\!\!(K) + \frac{(p-1)(q-1)}{2}\) under the technical condition \(q \geq (2 \nu^+\!\!(K) - 1)p - 1\). Moreover, if in addition \(n = \nu^+\!\!(K) = g_4\!(K)\), then \(\nu^+\!\!(K_{p,q}) = g_4\!(K_{p,q}) = p n + \frac{(p-1)(q-1)}{2}\). Under the negation of the above-mentioned condition, it is shown that \(\nu^+\!\!(K_{p,q}) \geq pq/2\) if \(\nu^+\!\!(K) > 0\).