Ozsvath-Szabo and Rasmussen invariants of doubled knots

الأصل · EN

Let νbe any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K, |ν(K)| <= g₄(K), and determines the 4-ball genus of positive torus knots, ν(Tₚ,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 ν(D₊(K,t)) = ± 1, then ν(D₋(K,t)) = 0. It is also shown that ν(D₊(K,t))= 1 for all t <= TB(K) and ν(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 × 2g Seifert matrix A and integer a, |a| <= g, there is a knot with Seifert form A and ν(K) = a.