Charles Livingston
Published 2006-02-27, updated 2006-03-11Version 2
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.
Comments: 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
Journal: Proc. Amer. Math. Soc. 136 (2008) 347-349
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Ozsvath-Szabo and Rasmussen invariants of cable knots
Ozsvath-Szabo and Rasmussen invariants of doubled knots
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