Slice knots with distinct Ozsváth-Szabó and Rasmussen invariants
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- by Charles Livingston
- Proc. Amer. Math. Soc. 136 (2008), 347-349
- DOI: https://doi.org/10.1090/S0002-9939-07-09276-3
- Published electronically: October 18, 2007
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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 $\mathbf {Z} \oplus \mathbf {Z} \oplus \mathbf {Z}$.References
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Bibliographic Information
- Charles Livingston
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 193092
- Email: livingst@indiana.edu
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 347-349
- MSC (2000): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9939-07-09276-3
- MathSciNet review: 2350422