Let $\nu$ be either the Ozsv\'ath-Szab\'o $\tau$-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. 查看全文>>