Inequality on $t_{\nu }(K)$ defined by Livingston and Naik and its applications

Abstract

Let $D_{+}(K,t)$ denote the positive $t$-twisted double of $K$. For a fixed integer-valued additive concordance invariant $\nu$ that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined $t_{\nu }(K)$ to be the greatest integer $t$ such that $\nu (D_{+}(K,t))=1$. Let $K_1$ and $K_2$ be any knots then we prove the following inequality : $t_{\nu }(K_1)+t_{\nu }(K_2)\le t_{\nu }(K_1\#K_2)\le min(t_{\nu }(K_1)-t_{\nu }(-K_2),t_{\nu }(K_2)-t_{\nu }(-K_1)).$ As an application we show that $t_{\tau }(K)\ne t_s(K)$ for infinitely many knots and that their difference can be arbitrarily large, where $t_{\tau }(K)$ (respectively $t_s(K)$) is $t_{\nu }(K)$ when $\nu$ is Ozváth-Szabó invariant $\tau$ (respectively when $\nu$ is normalized Rasmussen $s$ invariant).