Slice knots with distinct Ozsváth-Szabó and Rasmussen invariants

Livingston, Charles

doi:10.1090/S0002-9939-07-09276-3

Slice knots with distinct Ozsváth-Szabó and Rasmussen invariants
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Proc. Amer. Math. Soc. 136 (2008), 347-349 Request permission

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

A. J. Casson and C. McA. Gordon, Cobordism of classical knots, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 181–199. With an appendix by P. M. Gilmer. MR 900252
S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR 710056, DOI 10.4310/jdg/1214437665
Hisaaki Endo, Linear independence of topologically slice knots in the smooth cobordism group, Topology Appl. 63 (1995), no. 3, 257–262. MR 1334309, DOI 10.1016/0166-8641(94)00062-8
Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066
Patrick M. Gilmer, Slice knots in $S^{3}$, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135, 305–322. MR 711523, DOI 10.1093/qmath/34.3.305
Robert E. Gompf, Smooth concordance of topologically slice knots, Topology 25 (1986), no. 3, 353–373. MR 842430, DOI 10.1016/0040-9383(86)90049-2
M. Hedden, P. Ording, The Ozsváth-Szabó and Rasmussen concordance invariants are not equal (2005), arxiv.org/math/0512348.
Se-Goo Kim, Polynomial splittings of Casson-Gordon invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005), no. 1, 59–78. MR 2127228, DOI 10.1017/S0305004104008023
Charles Livingston, Splitting the concordance group of algebraically slice knots, Geom. Topol. 7 (2003), 641–643. MR 2026553, DOI 10.2140/gt.2003.7.641
Charles Livingston, Computations of the Ozsváth-Szabó knot concordance invariant, Geom. Topol. 8 (2004), 735–742. MR 2057779, DOI 10.2140/gt.2004.8.735
Charles Livingston and Swatee Naik, Ozsváth-Szabó and Rasmussen invariants of doubled knots, Algebr. Geom. Topol. 6 (2006), 651–657. MR 2240910, DOI 10.2140/agt.2006.6.651
C. Manolescu, B. Owens, A concordance invariant from the Floer homology of double branched covers (2005), arxiv.org/math.GT/0508065.
Peter Ozsváth and Zoltán Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639. MR 2026543, DOI 10.2140/gt.2003.7.615
J. A. Rasmussen, Khovanov homology and the slice genus (2003), arxiv.org/math/0306378.