INEQUALITY ON t(v)(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 v that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined t(v) (K) to be the greatest integer t such that v(D+(K,t)) = 1. Let K-1 and K-2 be any knots; then we prove the following inequality: t(v)(K-1) + t(v)(K-2) <= t(v)(K-1#K-2) <= min(t(v)(K-1) - t(v),(-K-2), t(v),(K-2) - t(v)(-K-1). As an application we show that l(T),-(K) not equal t(s)(K) for infinitely many knots and that their difference can be arbitrarily large, where t(tau)-(K) (respectively is t(s) (K))is t(v) (K) when v is an Ozvath-Szabo invariant tau (respectively when v is a normalized Rasmussen s invariant).