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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Slice knots with distinct Ozsváth-Szabó and Rasmussen invariants


Author: Charles Livingston
Journal: Proc. Amer. Math. Soc. 136 (2008), 347-349
MSC (2000): Primary 57M25
Published electronically: October 18, 2007
MathSciNet review: 2350422
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Abstract: As proved by Hedden and Ording, there exist knots for which the Ozsváth-Szabó 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 $ \mathbf{Z} \oplus \mathbf{Z}$. It thus follows quickly from the observation in this note that this concordance group contains a summand isomorphic to $ \mathbf{Z} \oplus \mathbf{Z} \oplus \mathbf{Z}$.


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Additional Information

Charles Livingston
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: livingst@indiana.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09276-3
PII: S 0002-9939(07)09276-3
Keywords: Slice knot, Ozsv\'ath-Szab\'o invariant, Rasmussen invariant, polynomial one
Received by editor(s): April 12, 2006
Published electronically: October 18, 2007
Additional Notes: The author’s research was supported by the NSF
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.