Ozsváth–Szabó and Rasmussen invariants of cable knots

Abstract

We study the behavior of the Ozsváth–Szabó and Rasmussen knot concordance invariants $\tau$ and $s$ on $K_{m,n}$, the $\left(m,n\right)$–cable of a knot $K$ where $m$ and $n$ are relatively prime. We show that for every knot $K$ and for any fixed positive integer $m$, both of the invariants evaluated on $K_{m,n}$ differ from their value on the torus knot $T_{m,n}$ by fixed constants for all but finitely many $n>0$. Combining this result together with Hedden’s extensive work on the behavior of $\tau$ on $\left(m,mr+1\right)$–cables yields bounds on the value of $\tau$ on any $\left(m,n\right)$–cable of $K$. In addition, several of Hedden’s obstructions for cables bounding complex curves are extended.