Concordance invariants of doubled knots and blowing up

Abstract:

Let $\nu$ be either the Ozsváth–Szabó $\tau$–invariant or the Rasmussen $s$–invariant, suitably normalized. For a knot $K$, Livingston and Naik defined the invariant $t_\nu (K)$ to be the minimum of $k$ for which $\nu$ of the $k$–twisted positive Whitehead double of $K$ vanishes. They proved that $t_\nu (K)$ is bounded above by $-TB(-K)$, where $TB$ is the maximal Thurston–Bennequin number. We use a blowing-up process to find a crossing change formula and a new upper bound for $t_\nu$ in terms of the unknotting number. As an application, we present infinitely many knots $K$ such that the difference between Livingston–Naik’s upper bound $-TB(-K)$ and $t_\nu (K)$ can be arbitrarily large.