Concordance invariants of doubled knots and blowing up

Let ν be either the Ozsváth–Szabó τ –invariant or the Rasmussen s–invariant, suitably normalized. For a knot K, Livingston and Naik defined the invariant t _ν (K) to be the minimum of k for which ν of the k–twisted positive Whitehead double of K vanishes. They proved that t _ν (K) is bounded above by −T B(−K), where T B 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 _ν 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 −T B(−K) and t _ν (K) can be arbitrarily large.