Slice knots with distinct Ozsvath-Szabo and Rasmussen Invariants
Abstract
Description
As proved by Hedden and Ording, there exist knots for which the Ozsvath-Szabo and Rasmussen smooth concordance invariants, tau 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 tau and s differ. Manolescu and Owens have previously found a concordance invariant that is independent of both tau and 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 a free abelian group on two generators. It thus follows quickly from the observation in this note that this concordance group contains a subgroup isomorphic to a free abelian group on three generators.
In the first version of this note, the main result was applied to show that the smooth concordance group of topologically slice knots contains a summand isomorphic to a free abelian group on two generators. The author has learned that Manolescu and Owens previously found a knot invariant, based on the Heegaard-Floer Homology of the 2-fold branched cover of the knot, that they used to detect such a summand (see math.GT/0508065). Thus, a consequence of this note is that the work of Hedden-Ording and Manolescu-Owens combines to give a summand that is free on three generators
In the first version of this note, the main result was applied to show that the smooth concordance group of topologically slice knots contains a summand isomorphic to a free abelian group on two generators. The author has learned that Manolescu and Owens previously found a knot invariant, based on the Heegaard-Floer Homology of the 2-fold branched cover of the knot, that they used to detect such a summand (see math.GT/0508065). Thus, a consequence of this note is that the work of Hedden-Ording and Manolescu-Owens combines to give a summand that is free on three generators