Slice knots with distinct Ozsváth-Szabó and Rasmussen invariants

Abstract

As proved by Hedden and Ording, there exist knots for which the Ozsváth-Szabó and Rasmussen smooth concordance invariants, τ \tau and s 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 \tau and s s differ. Manolescu and Owens have previously found a concordance invariant that is independent of both τ \tau and s s on knots of polynomial one, and as a consequence have shown that the smooth concordance group of topologically slice knots contains a summand isomorphic to Z ⊕ Z \mathbf {Z} \oplus \mathbf {Z} . It thus follows quickly from the observation in this note that this concordance group contains a summand isomorphic to Z ⊕ Z ⊕ Z \mathbf {Z} \oplus \mathbf {Z} \oplus \mathbf {Z} .