Ozsváth–Szabó and Rasmussen invariants of doubled knots

Abstract

Let \nu be any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K, |\nu(K)| <= g_4(K), and determines the 4-ball genus of positive torus knots, \nu(T_{p,q}) = (p-1)(q-1)/2. Either of the knot concordance invariants of Ozsvath-Szabo or Rasmussen, suitably normalized, have these properties. Let D_{\pm}(K,t) denote the positive or negative t-twisted double of K. We prove that if \nu(D_{+}(K,t)) = \pm 1, then \nu(D_{-}(K,t)) = 0. It is also shown that \nu(D_{+}(K,t))= 1 for all t <= TB(K) and \nu(D_{+}(K, t)) = 0 for all t \ge -TB(-K), where TB(K) denotes the Thurston-Bennequin number. A realization result is also presented: for any 2g \times 2g Seifert matrix A and integer a, |a| <= g, there is a knot with Seifert form A and \nu(K) = a.