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Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
     

Slice knots with distinct Ozsváth-Szabó and Rasmussen invariants

Author(s): Charles Livingston
Journal: Proc. Amer. Math. Soc. 136 (2008), 347-349.
MSC (2000): Primary 57M25
Posted: 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:

[CG]
A. Casson, C. McA. Gordon, Cobordism of classical knots, in À la recherche de la topologie perdue, ed. by Guillou and Marin, Progress in Mathematics, Volume 62, 1986. (Originally published as an Orsay Preprint, 1975.) MR 900252

[D]
S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279-315. MR 710056 (85c:57015)

[E]
H. Endo, Linear independence of topologically slice knots in the smooth cobordism group, Topology Appl. 63 (1995), no. 3, 257-262. MR 1334309 (96c:57010)

[F]
M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453. MR 679066 (84b:57006)

[Gi]
P. Gilmer, Slice knots in $ S\sp{3}$, Quart. J. Math. Oxford Ser. (2) 34 (1983), 305-322. MR 711523 (85d:57004)

[Go]
R. Gompf, Smooth concordance of topologically slice knots, Topology 25 (1986), no. 3, 353-373. MR 842430 (87i:57004)

[HO]
M. Hedden, P. Ording, The Ozsváth-Szabó and Rasmussen concordance invariants are not equal (2005), arxiv.org/math/0512348.

[K]
S.-G. Kim, Polynomial splittings of Casson-Gordon invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005), 59-78. MR 2127228 (2005m:57018)

[L1]
C. Livingston, Splitting the concordance group of algebraically slice knots, Geom. Topol. 7 (2003), 641-643. MR 2026553 (2004j:57009)

[L2]
C. Livingston, Computations of the Ozsváth-Szabó knot concordance invariant, Geom. Topol. 8 (2004), 735-742. MR 2057779 (2005d:57019)

[LN]
C. Livingston, S. Naik, Ozsváth-Szabó and Rasmussen invariants of doubled knots, Algebr. Geom. Topol. 6 (2006), 651-657. MR 2240910 (2007d:57023)

[MO]
C. Manolescu, B. Owens, A concordance invariant from the Floer homology of double branched covers (2005), arxiv.org/math.GT/0508065.

[OS]
P. Ozsváth, Z. Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615-639. MR 2026543 (2004i:57036)

[Ra]
J. A. Rasmussen, Khovanov homology and the slice genus (2003), arxiv.org/math/0306378.


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Additional Information:

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

DOI: 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
Posted: October 18, 2007
Additional Notes: The author's research was supported by the NSF
Communicated by: Daniel Ruberman
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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