Slice Knots with Distinct Ozsváth-szabó and Rasmussen Invariants

As proved by Hedden and Ording, there exist knots for which the Ozsváth-Szabó and Rasmussen smooth concordance invariants, τ and s, differ. The Hedden-Ording examples have nontrivial Alexander polynomials and are not topologically slice. It is shown in this note that a simple manipulation of the Hedden-Ording examples yields a topologically slice Alexander polynomial one knot for which τ and s differ. It follows that the smooth concordance group of topologically slice knots contains a summand isomorphic to Z⊕ Z. The Ozsváth-Szabó and Rasmussen knot concordance invariants [OS, Ra], τ and s, are each powerful invariants, sufficient to resolve the Milnor conjecture on the 4-genus of torus knots. It had been conjectured that τ = s/2, but recently Hedden and Ording [HO] provided a counterexample, showing that certain doubled knots, including D+(T2,3, 2) and D+(T2,5, 4), satisfy τ = 0 and s = 2. Here D+(Tp,q, t) denotes the t-twisted positive double of the (p, q)-torus knot. These examples demonstrate the richness of these new invariants. However, in and of themselves, they do not reveal any new structure of the concordance group. For instance, although some of these examples are algebraically slice, all can be shown not to be even topologically slice using Casson-Gordon invariants [CG]. (See [G] for techniques that resolve the nonsliceness of these particular doubled knots, and [K] for a general discussion of Casson-Gordon invariants and doubled knots.) In this note it will be shown that the basic Hedden-Ording examples can be manipulated to yield a knot with Alexander polynomial one (and thus, by Freedman [F], a topologically slice knot) for which τ and s/2 differ. The smooth concordance group contains a subgroup S consisting of topologically slice knots. In [L2] it was already shown that τ and s/2 agree and are nonzero on some polynomial one knots, such as D+(T2,3, 0). Hence τ and s/2 are linearly independent integer valued homomorphisms on S, and it follows that S contains a summand isomorphic to Z⊕ Z. This expands on the Z summand found in [L1]. 1. An Alexander polynomial one knot with τ 6= s/2. The doubled knots D+(K, t) bounds a Seifert surfaces built by adding two bands to a disk, one with framing −1 and the other with framing t. With respect to the corresponding basis of the first homology of the surface, the Seifert matrix is: