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Links not concordant to the Hopf link

Published online by Cambridge University Press:  24 February 2014

STEFAN FRIEDL  and
MARK POWELL
STEFAN FRIEDL
Affiliation:
Fakultät für Mathematik, Universität Regensburg, Germany. e-mail: sfriedl@gmail.com
MARK POWELL
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana, U.S.A. e-mail: macp@indiana.edu
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Abstract

We give new Casson–Gordon style obstructions for a two–component link to be topologically concordant to the Hopf link.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 3 , May 2014 , pp. 425 - 459
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[CF10]Cha, J. C. and Friedl, S. Twisted torsion invariants and link concordance. To appear in Forum Math., Preprint: http://arxiv.org/abs/1001.0926, (2010).Google Scholar
[CG86]Casson, A. and Gordon, C. McA.Cobordism of classical knots. In A la Recherche de la Topologie Perdue Progr. Math. vol. 62 (Birkhauser Boston, 1986), pages 181–199.Google Scholar
[Cha07]Cha, J. C.The structure of the rational concordance group of knots. Mem. Amer. Math. Soc. 189 (885) (2007). x+95.Google Scholar
[Cha10]Cha, J. C.Link concordance, homology cobordism, and Hirzebruch-type defects from iterated p-covers. J. Eur. Math. Soc. 12 (2010), 555–610.CrossRefGoogle Scholar
[Cha12]Cha, J. C. Symmetric Whitney tower cobordism for bordered 3–manifolds and links. arXiv:1204.4968 [math.GT] (2012).Google Scholar
[CK99a]Cha, J. C. and Ko, K. H.On equivariant slice knots. Proc. Amer. Math. Soc. 127 (7) (1999), 2175–2182.Google Scholar
[CK99b]Cha, J. C. and Ko, K. H.Signature invariants of links from irregular covers and non-abelian covers. Math. Proc. Camb. Philos. Soc. 127 (1) (1999), 67–81.CrossRefGoogle Scholar
[CK02]Cha, J. C. and Ko, K. H.Signatures of links in rational homology spheres. Topology 41 (2002), 1161–1182.Google Scholar
[CK06]Cha, J. C. and Ko, K. H.Signature invariants of covering links. Trans. Amer. Math. Soc. 358 (2006), 3399–3412.Google Scholar
[CK08]Cha, J. C. and Kim, T.Covering link calculus and iterated Bing doubles. Geom. Topol. 12 (2008), 2172–2201.Google Scholar
[CKRS10]Cha, J. C., Kim, T., Ruberman, D. and Strle, S. Smooth concordance of links topologically concordant to the Hopf link. arXiv:1012.2045v1[math.GT] (2010).Google Scholar
[CLR08]Cha, J. C., Livingston, C. and Ruberman, D.Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles. Math. Proc. Camb. Phil. Soc. 144 (2008), 403–410.Google Scholar
[CO90]Cochran, T. D. and Orr, K. E.Not all links are concordant to boundary links. Bulletin Amer. Math. Soc. 23 (1) (1990), 99–106.CrossRefGoogle Scholar
[CO93]Cochran, T. D. and Orr, K. E.Not all links are concordant to boundary links. Ann. Math. 138 (1993), 519–554.Google Scholar
[CO09]Cha, J. C. and Orr, K. E.L(2)-signatures, homology localization and amenable groups. To appear in Comm. Pure Appl. Math. Preprint: arXiv:09103.3700 (2009).Google Scholar
[COT03]Cochran, T. D., Orr, K. E. and Teichner, P.Knot concordance, Whitney towers and L (2) signatures. Ann. of Math. (2) 157 (2003), no. 2, 433–519.Google Scholar
[CST12]Conant, J., Schneiderman, R. and Teichner, P.Whitney tower concordance of classical links. Geom. Topol. 16 (2012), 1419–1479.Google Scholar
[Dav06]Davis, J.A two component link with Alexander polynomial one is concordant to the Hopf link. Math. Proc. Camb. Phil. Soc. 140 (2006), no. 2, 265–268.Google Scholar
[Fox56]Fox, R. H.Free differential calculus. III. Subgroups. Ann. of Math. (2) 64 (1956), 407–419.Google Scholar
[FP12]Friedl, S. and Powell, M.An injectivity theorem for Casson–Gordon type representations relating to the concordance of knots and links. Bull. Korean Math. Soc. 49 (2012), 395–409.Google Scholar
[FQ90]Freedman, M. and Quinn, F.Topology of Four Manifolds (Princeton University Press, 1990).CrossRefGoogle Scholar
[Fre84]Freedman, M. H.The disk theorem for four-dimensional manifolds. Proc. Internat. Congr. Math. vol. 1, 2 (Warsaw, 1983), PWN (1984), pages 647–663.Google Scholar
[Fri03]Friedl, S.Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants. PhD thesis, Brandeis University (2003).CrossRefGoogle Scholar
[Fri04]Friedl, S.Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants. Alg. Geom. Topol. 4 (2004), 893–934.CrossRefGoogle Scholar
[Fri05]Friedl, S.Link concordance, boundary link concordance and eta-invariants. Math. Proc. Camb. Phil. Soc. 138 (2005), no. 3, 437–460.CrossRefGoogle Scholar
[Har08]Harvey, S.Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group. Geom. Topol. 12 (2008), no. 1, 387–430.Google Scholar
[Hil02]Hillman, J. A.Algebraic Invariants of Links. Series on Knots and Everything 32 (World Scientific Publishing Co. 2002).Google Scholar
[HS97]Hillman, J. A. and Sakuma, M.On the homology of finite abelian coverings of links. Canad. Math. Bull. 40 (1997), 309–315.Google Scholar
[Kaw77]Kawauchi, A.On quadratic forms of 3-manifolds. Invent. Math. 43 (1977), 177–198.CrossRefGoogle Scholar
[Kaw78]Kawauchi, A.On the Alexander polynomials of cobordant links. Osaka J. Math. 15 (1978), no. 1, 151–159.Google Scholar
[Let00]Letsche, C. F.An obstruction to slicing knots using the eta invariant. Math. Proc. Camb. Phil. Soc. 128 (2) (2000), 301–319.Google Scholar
[Lev69]Levine, J.Knot cobordism groups in codimension two. Comment. Math. Helv. 44 (1969), 229–244.Google Scholar
[Lev94]Levine, J.Link invariants via the eta invariant. Comment. Math. Helv. 69 (1994), 82–119.Google Scholar
[Lev07]Levine, J.Concordance of boundary links. J. Knot Theory Ramifications 16 (2007), no. 9, 1111–1120.CrossRefGoogle Scholar
[Lit84]Litherland, R. A. Cobordism of satellite knots. In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., (Amer. Math. Soc. 1984) pages 327–362.Google Scholar
[Mil57]Milnor, J. W. Isotopy of links. Alg. Geom. Top. A symposium in honor of S. Lefschetz, (1957), 280–306.Google Scholar
[MM82]Mayberry, J. P. and Murasugi, K.Torsion-groups of abelian coverings of links. Trans. Amer. Math. Soc. 271 (1982), 143–173.CrossRefGoogle Scholar
[Mur67]Murasugi, K.On a certain numerical invariant of link types. Trans. Amer. Math. Soc. 117 (1967), 387–422.Google Scholar
[Nak78]Nakagawa, Y.On the Alexander polynomials of slice links. Osaka J. Math. 15 (1978), no. 1, 161–182.Google Scholar
[Por04]Porti, J.Mayberry–Murasugi's formula for links in homology 3-spheres. Proc. Amer. Math. Soc. 132 (2004), 3423–3431.Google Scholar
[Sak95]Sakuma, M.Homology of abelian coverings of links and spatial graphs. Canad. J. Math. 47 (1995), 201–224.Google Scholar
[Smo89]Smolinsky, L.Invariants of link cobordism. Proceedings of the 1987 Georgia Topology Conference (Athens, GA, 1987). Topology Appl. 32 (1989), 161–168.Google Scholar
[Sta65]Stallings, J.Homology and central series of groups. J. Algebra 2 (1965), 170–181.CrossRefGoogle Scholar
[Tri69]Tristram, A. G.Some cobordism invariants for links. Proc. Camb. Phil. Soc. 66 (1969), 251–264.Google Scholar
[Tur86]Turaev, V.Reidemeister torsion in knot theory. Russian Math. Surveys 41 (1986), 119–182.CrossRefGoogle Scholar
[Web79]Weber, C.Sur une formule de R. H. Fox concernant l'homologie des revêtements cycliques. Enseign. Math. (2) 25 (1979), 261–272.Google Scholar