Ozsvath-Szabo and Rasmussen invariants of doubled knots

Let nu be any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K, vertical bar nu(K)vertical bar <= g(4) (K), and determines the 4-ball genus of positive torus knots, nu(T-p,T-q) = (p - 1) (q - 1)/2. Either of the knot concordance invariants of Ozsvath-Szabo or Rasmussen, suitably normalized, have these properties. Let D +/- (K,t) denote the positive or negative t-twisted double of K. We prove that if nu( D (+) (K, t)) = +/- 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 >= -TB (-K), where TB(K) denotes the Thurston-Bennequin number. A realization result is also presented: for any 2g x 2g Seifert matrix A and integer a, vertical bar a vertical bar <= g, there is a knot with Seifert form A and nu(K) = a.