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Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles

Published online by Cambridge University Press:  01 March 2008

JAE CHOON CHA ,
CHARLES LIVINGSTON  and
DANIEL RUBERMAN
JAE CHOON CHA
Affiliation:
Department of Mathematics Pohang University of Science and Technology (POSTECH) Pohang, Kyungbok 790-784Republic of Korea. e-mail: [email protected]
CHARLES LIVINGSTON
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405, U.S.A. e-mail: [email protected]
DANIEL RUBERMAN
Affiliation:
Department of Mathematics, MS 050, Brandeis University, Waltham, Massachusetts 02454, U.S.A. e-mail: [email protected]
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Abstract

If the Bing double of a knot K is slice, then K is algebraically slice. In addition the Heegaard–Floer concordance invariants τ, developed by Ozsváth–Szabó, and δ, developed by Manolescu and Owens, vanish on K.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 2 , March 2008 , pp. 403 - 410
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Beliakova, A. and Wehrli, A.. Categorification of the colored Jones polynomial and Rasmussen invariant of links, arxiv.org/math.QA/0510382.Google Scholar
[2]Cha, J. C. Link concordance, homology cobordism, and Hirzebruch-type intersection form defects from towers of iterated p-covers, arxiv:0705.0088.Google Scholar
[3]Cha, J. C.The structure of the rational concordance group of knots. Mem. Amer. Math. Soc. 189 (2007), no.885.Google Scholar
[4]Cha, J. C. and Ko, K. H.. Signature invariants of covering links. Trans. Amer. Math. Soc. 358 (2006), 3399–3412.CrossRefGoogle Scholar
[5]Cochran, T. and Orr, K.. Not all links are concordant to boundary links. Ann. of Math. (2) 138 (1993), 519–554.CrossRefGoogle Scholar
[6]Cimasoni, D.Slicing Bing doubles. Alg. Geom. Topol. 6 (2006) 2395–2415, msp.warwick.ac.uk/agt/2006/06/p083.xhtml.CrossRefGoogle Scholar
[7]Habiro, K.Claspers and finite type invariants of links. Geom. Topol. 4 (2000), 1–83.CrossRefGoogle Scholar
[8]Harvey, S. Homology cobordism invariants and the Cochran-Orr-Teichner filtration of the link concordance group, arxiv.org/math.GT/0609378.Google Scholar
[9]Harvey, S. New phenomena in knot and link concordance (joint work with Tim Cochran and Constance Leidy); Oberwolfach Reports 3, to appear, mfo.de/programme/schedule/2006/32/OWR_2006_35.pdf.Google Scholar
[10]Kawauchi, A.On links not cobordant to split links. Topology 19 (1980), 321–334.CrossRefGoogle Scholar
[11]Kearton, C.The Milnor signatures of compound knots. Proc. Amer. Math. Soc. 76 (1979), 157–160.CrossRefGoogle Scholar
[12]Krushkal, V.Exponential separation in 4-manifolds. Geom. Topol. 4 (2000), 397–405.CrossRefGoogle Scholar
[13]Krushkal, V.On the relative slice problem and four-dimensional topological surgery. Math. Ann. 315 (1999), 363–396.CrossRefGoogle Scholar
[14]Krushkal, V.A counterexample to the strong version of Freedman's conjecture, arxiv.org/math. GT/0610865.Google Scholar
[15]Krushkal, V. and Teichner, P.. Alexander duality, gropes and link homotopy. Geom. Topol. 1 (1997), 51–69.CrossRefGoogle Scholar
[16]Levine, J.Knot cobordism groups in codimension two. Comment. Math. Helv. 44 (1969) 229–244.CrossRefGoogle Scholar
[17]Levine, J.Invariants of knot cobordism. Invent. Math. 8 (1969), 98–110.CrossRefGoogle Scholar
[18]Livingston, C.Computations of the Ozsváth–Szabó knot concordance invariant. Geom. Topol. 8 (2004), 735–742.CrossRefGoogle Scholar
[19]Livingston, C. and Melvin, P. Abelian invariants of satellite knots, geometry and topology (College Park, Md., 1983/84). 217–227. Lecture Notes in Math. 1167 (Springer, 1985).CrossRefGoogle Scholar
[20]Manolescu, C. and Owens, B.. A concordance invariant from the Floer homology of double branched covers, arxiv.org/math.GT/0508065.Google Scholar
[21]Ozsváth, P. and Szabó, Z.. Knot Floer homology and the four-ball genus. Geom. Topol. 7 (2003), 615–639.CrossRefGoogle Scholar
[22]Rasmussen, J. A. Khovanov homology and the slice genus. To appear in Invent. Math., arxiv.org/math/0306378.Google Scholar
[23]Schneiderman, R. and Teichner, P.. Whitney towers and the Kontsevich integral. Proceedings of the Casson Fest Geom. Topol. Monogr. 7 (2004) 101–134.CrossRefGoogle Scholar
[24]Seifert, H.On the homology invariants of knots. Quart. J. Math. Oxford Ser. (2) 1 (1950), 23–32.CrossRefGoogle Scholar