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Concordance of Bing Doubles and Boundary Genus†

Published online by Cambridge University Press:  18 July 2011

CHARLES LIVINGSTON  and
CORNELIA A. VAN COTT
CHARLES LIVINGSTON
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN, 47405, U.S.A. e-mail: [email protected]
CORNELIA A. VAN COTT
Affiliation:
Department of Mathematics, University of San Francisco, San Francisco, CA, 94117, U.S.A. e-mail: [email protected]
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Abstract

Cha and Kim proved that if a knot K is not algebraically slice, then no iterated Bing double of K is concordant to the unlink. We prove that if K has nontrivial signature σ, then the n–iterated Bing double of K is not concordant to any boundary link with boundary surfaces of genus less than 2n−1σ. The same result holds with σ replaced by 2τ, twice the Ozsváth–Szabó knot concordance invariant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 3 , November 2011 , pp. 459 - 470
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Bing, R. H.A homeomorphism between the 3-sphere and the sum of two solid horned spheres. Ann. of Math. (2) 56 (1952), 354–362.CrossRefGoogle Scholar
[2]Cappell, S. and Shaneson, J.Link cobordism. Comment. Math. Helv. 55 (1980), 20–49.CrossRefGoogle Scholar
[3]Casson, A. J. and Gordon, C. McA. Cobordism of classical knots, with an appendix by P. M. Gilmer, in Progr. Math. 62, ‘Á la recherche de la topologie perdue,’ 181–199 (Birkhäuser Boston, Boston, MA, 1986).Google Scholar
[4]Cha, J. C.Link concordance, homology cobordism and Hirzebruch-type defects from iterated p–covers. J. Eur. Math. Soc. (JEMS) 12 (2010), 555–610.CrossRefGoogle Scholar
[5]Cha, J. C. and Kim, T.Covering link calculus and iterated Bing doubles. Geom. Topol. 12 (2008), 2173–2201.CrossRefGoogle Scholar
[6]Cha, J. C., Livingston, C. and Ruberman, D.Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles. Math. Proc. of the Camb. Phil. Soc. 144 (2008), 403–410.CrossRefGoogle Scholar
[7]Cimasoni, D.Slicing Bing doubles. Algebr. Geom. Topol. 6 (2006), 2395–2415.Google Scholar
[8]Cochran, T., Harvey, S. and Leidy, C.Link concordance and generalized doubling operators. Algebr. Geom. Topology 8 (2008), 1593–1646.Google Scholar
[9]Cochran, T. and Orr, K.Not all links are concordant to boundary links. Ann. of Math. (2) 138 (1993), 519–554.CrossRefGoogle Scholar
[10]Conner, P. and Floyd, E.Differentiable periodic maps. Ergeb. Math. Grenzgeb., N. F., Band 33 (Academic Press Inc., Springer-Verlag, 1964).Google Scholar
[11]Freedman, M. The disk theorem for four-dimensional manifolds. Proc. I.C.M. (Warsaw), (1983), 647–663.Google Scholar
[12]Freedman, M. and Quinn, F.Topology of 4–manifolds, (Princeton University Press, 1990).Google Scholar
[13]Gilmer, P. and Livingston, C.The Casson-Gordon invariant and link concordance. Topology 31 (1992), 475–492.Google Scholar
[14]Harvey, S.Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group. Geom. Topol. 12 (2008), 387–430.Google Scholar
[15]Levine, A. Slicing mixed Bing-Whitehead doubles. arxiv.org/abs/0912.5222.Google Scholar
[16]Levine, J.Invariants of knot cobordism. Invent. Math. 8 (1969), 98–110.CrossRefGoogle Scholar
[17]Levine, J.Link invariants via the eta invariant. Comment. Math. Helv. 69 (1994), 82–119.CrossRefGoogle Scholar
[18]Litherland, R.Cobordism of satellite knots, Four-manifold theory (Durham, N.H., 1982), 327–362; Contemp. Math. 35 (Amer. Math. Soc., 1984).Google Scholar
[19]Livingston, C.Links not concordant to boundary links. Proc. Amer. Math. Soc. 110 (1990), 1129–1131.Google Scholar
[20]Livingston, C.Computations of the Ozsváth–Szabó knot concordance invariant. Geom. Topol. 8 (2004), 735–742.CrossRefGoogle Scholar
[21]Murakami, H. and Yasuhara, A.Four-genus and four-dimensional clasp number of a knot, Proc. Amer. Math. Soc. 128 (2000), 3693–3699.CrossRefGoogle Scholar
[22]Murasugi, K.On a certain numerical invariant of link types. Trans. Amer. Math. Soc. 117 (1965), 387–422.CrossRefGoogle Scholar
[23]Ozsváth, P. and Szabó, Z.Knot Floer homology and the four-ball genus. Geom. Topol. 7 (2003), 615–639.Google Scholar
[24]Plamenevskaya, O.Bounds for Thurston-Bennequin number from Floer homology. Algebr. Geom. Topol. 4 (2004), 399–406.CrossRefGoogle Scholar
[25]Rasmussen, J.Khovanov homology and the slice genus. Invent. math. 182 (2010), 419–447.CrossRefGoogle Scholar
[26]Rolfsen, D., Knots and Links, Mathematics Lecture Series, 7 (Publish or Perish, 1990).Google Scholar
[27]Teichner, P., unpublished.Google Scholar
[28]Tristram, A. G.Some cobordism invariants for links. Proc. Camb. Phil. Soc. 66 (1969), 251–264.CrossRefGoogle Scholar
[29]Van Cott, C. An obstruction to slicing iterated Bing doubles. arxiv.org/abs/0907.4948.Google Scholar